Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.4s

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: 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

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (*
    (log
     (-
      (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u) 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. Final simplification 99.1%

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

Alternative 2: 97.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     (/ 1.0 u)
     (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
    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 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 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 8.3% accurate, 1.8× speedup

Math

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

FPCore

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 3.3

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

5. Applied rewrites 3.3%

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

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 12.4%

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

9. Final simplification 12.1%

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

Alternative 4: 24.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{PI}\left(\right)\\ \log \left(\left(0.5 \cdot \frac{\mathsf{PI}\left(\right)}{s} + \left(\frac{t\_0 \cdot -0.5 - t\_0 \cdot u}{s} + 0.5\right)\right) \cdot 4 - 1\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

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

4. Applied rewrites 97.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-2} - {\left(\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u\right)}^{2}}{e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)} - \left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u}}} - 1\right) \]

5. Taylor expanded in s around inf

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

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

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

   2. distribute-lft-out N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-out N/A

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

7. Applied rewrites 25.1%

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

8. Final simplification 25.1%

   \[\leadsto \log \left(\left(0.5 \cdot \frac{\mathsf{PI}\left(\right)}{s} + \left(\frac{\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot -0.5 - \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot u}{s} + 0.5\right)\right) \cdot 4 - 1\right) \cdot \left(-s\right) \]
9. Add Preprocessing

Alternative 5: 25.0% accurate, 3.8× speedup

Math

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

FPCore

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

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

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

5. Applied rewrites 13.8%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 17.1%

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

9. Taylor expanded in s around inf

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

11. Applied rewrites 25.1%

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

12. Final simplification 25.1%

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

Alternative 6: 14.0% 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 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 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 98.3%

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

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 11.2

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

8. Applied rewrites 11.2%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]
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. lower-/.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 13.7

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

10. Applied rewrites 13.7%

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

11. Final simplification 13.7%

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

Alternative 7: 7.4% accurate, 23.2× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (* -4.0 (fma (* -0.5 (PI)) u (* 0.25 (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 s around -inf

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

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

5. Applied rewrites 14.2%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 16.6%

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

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. *-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. *-commutative N/A

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

    9. lower-fma.f32 N/A

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

    10. lower-*.f32 N/A

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

    11. lower-PI.f32 N/A

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

    12. lower-*.f32 N/A

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

    13. lower-PI.f32 11.2

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

11. Applied rewrites 11.2%

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

12. Final simplification 11.2%

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

Alternative 8: 11.5% 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 11.2

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

5. Applied rewrites 11.2%

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

Alternative 9: 10.3% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

4. Applied rewrites 97.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-2} - {\left(\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u\right)}^{2}}{e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)} - \left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u}}} - 1\right) \]

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. exp-neg N/A

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

   5. rem-exp-log N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 10.2

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

7. Applied rewrites 10.2%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 10.2%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024285 
(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))))