Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.2s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (+
      (/ 1.0 (+ (exp (/ (* (PI) s) (* s s))) 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. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. neg-sub0 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 - \left(-\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

   5. div-sub N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0}{s} - \frac{-\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

   6. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(s\right)}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0}}{\mathsf{neg}\left(s\right)} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   8. lift-neg.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0}{\color{blue}{-s}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   9. frac-sub N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   11. lower--.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}} - 1\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

   14. lower-*.f32 98.9

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-s\right) \cdot s}}}}} - 1\right) \]

4. Applied rewrites 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   2. frac-2neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}}}}} - 1\right) \]

   3. distribute-frac-neg N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}\right)}}}} - 1\right) \]

   4. lift--.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\mathsf{neg}\left(\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}\right)}}} - 1\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\mathsf{neg}\left(\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}\right)}}} - 1\right) \]

   6. mul0-lft N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\mathsf{neg}\left(\frac{\color{blue}{0} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(\left(-s\right) \cdot s\right)}\right)}}} - 1\right) \]

   7. neg-sub0 N/A

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

   8. frac-2neg N/A

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

   9. distribute-frac-neg2 N/A

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

   10. lower-/.f32 N/A

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 2: 16.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{-1}{e^{t\_0} + 1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, -4 \cdot \left(\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 2\right) - -2 \cdot t\_0\right) - 1\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

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))
        -9.9999998245167e-14)
     (fma (/ (* (* (/ 0.0 u) u) u) s) -0.5 (* -4.0 (* (fma -0.5 u 0.25) (PI))))
     (*
      (log
       (-
        (-
         (fma (/ (fma (* 0.5 (PI)) u (* 0.25 (PI))) s) -4.0 2.0)
         (* -2.0 t_0))
        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)))) < -9.99999982e-14

   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

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

   5. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 7.8%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \left(-\left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

   8. Taylor expanded in u around -inf

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

   10. Applied rewrites 16.5%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{0}{u} + 0\right) \cdot u\right) \cdot u}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 7.2%

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

   if -9.99999982e-14 < (*.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. 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. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 - \left(-\mathsf{PI}\left(\right)\right)}}{s}}}} - 1\right) \]

      5. div-sub N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0}{s} - \frac{-\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{\mathsf{neg}\left(0\right)}{\mathsf{neg}\left(s\right)}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0}}{\mathsf{neg}\left(s\right)} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      8. lift-neg.f32 N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0}{\color{blue}{-s}} - \frac{-\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

      9. frac-sub N/A

         \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      10. lower-/.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

      11. lower--.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\color{blue}{0 \cdot s} - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}} - 1\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \color{blue}{\left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}}{\left(-s\right) \cdot s}}}} - 1\right) \]

      14. lower-*.f32 98.8

          \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   4. Applied rewrites 98.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\color{blue}{\frac{0 \cdot s - \left(-s\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}{\left(-s\right) \cdot s}}}}} - 1\right) \]

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in s around inf

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

      1. lower--.f32 N/A

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

   9. Applied rewrites 20.9%

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

4. Final simplification 16.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}}\right) \cdot \left(-s\right) \leq -9.9999998245167 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, -4 \cdot \left(\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -4, 2\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) - 1\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

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: 97.6% 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 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 98.0%

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

6. Final simplification 98.0%

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

Alternative 5: 7.4% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (* (PI) (PI)) s)))
   (fma
    (*
     (* u u)
     (fma -4.0 (+ (/ t_0 u) t_0) (* (fma (/ (PI) u) (/ (PI) s) t_0) 4.0)))
    -0.5
    (* (fma (* -0.5 u) (PI) (* 0.25 (PI))) -4.0))))

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

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

5. Applied rewrites 8.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]
6. Step-by-step derivation

7. Applied rewrites 10.4%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(-0.5 \cdot u, \mathsf{PI}\left(\right), 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

8. Taylor expanded in u around inf

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

10. Applied rewrites 11.0%

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

11. Final simplification 11.0%

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

Alternative 6: 7.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma (/ (* (* (/ 0.0 u) u) u) s) -0.5 (* (- (* 2.0 (PI)) (/ (PI) u)) u)))

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

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

5. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]
6. Step-by-step derivation

7. Applied rewrites 7.3%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \left(-\left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

8. Taylor expanded in u around -inf

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

10. Applied rewrites 11.0%

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

11. Taylor expanded in u around inf

    \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{0}{u} + 0\right) \cdot u\right) \cdot u}{s}, \frac{-1}{2}, u \cdot \left(-1 \cdot \frac{\mathsf{PI}\left(\right)}{u} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]
12. Step-by-step derivation

13. Applied rewrites 11.3%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{0}{u} + 0\right) \cdot u\right) \cdot u}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\right) \]

14. Final simplification 11.3%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\right) \]
15. Add Preprocessing

Alternative 7: 7.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, \left(\left(-0.5 \cdot u\right) \cdot \mathsf{PI}\left(\right) + 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma
  (/ (* (* (/ 0.0 u) u) u) s)
  -0.5
  (* (+ (* (* -0.5 u) (PI)) (* 0.25 (PI))) -4.0)))

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

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

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]
6. Step-by-step derivation

7. Applied rewrites 7.3%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \left(-\left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

8. Taylor expanded in u around -inf

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

10. Applied rewrites 11.0%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{0}{u} + 0\right) \cdot u\right) \cdot u}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]
11. Step-by-step derivation

12. Applied rewrites 11.3%

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

13. Final simplification 11.3%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, \left(\left(-0.5 \cdot u\right) \cdot \mathsf{PI}\left(\right) + 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]
14. Add Preprocessing

Alternative 8: 7.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, -4 \cdot \left(\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma (/ (* (* (/ 0.0 u) u) u) s) -0.5 (* -4.0 (* (fma -0.5 u 0.25) (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 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 8.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]
6. Step-by-step derivation

7. Applied rewrites 7.9%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \left(-\left(-\mathsf{PI}\left(\right)\right)\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

8. Taylor expanded in u around -inf

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

10. Applied rewrites 11.0%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{0}{u} + 0\right) \cdot u\right) \cdot u}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]
11. Step-by-step derivation

12. Applied rewrites 7.0%

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

13. Final simplification 11.0%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{0}{u} \cdot u\right) \cdot u}{s}, -0.5, -4 \cdot \left(\mathsf{fma}\left(-0.5, u, 0.25\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]
14. Add Preprocessing

Alternative 9: 11.6% accurate, 11.9× speedup

Math

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

FPCore

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

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

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

5. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]

6. Taylor expanded in s around 0

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

8. Applied rewrites 10.1%

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

9. Taylor expanded in u around -inf

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

11. Applied rewrites 11.3%

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

12. Final simplification 11.3%

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

Alternative 10: 11.4% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

Alternative 11: 10.2% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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

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

5. Applied rewrites 8.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)} \]

6. Taylor expanded in s around 0

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

8. Applied rewrites 10.1%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 10.1%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(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))))