Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.5s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\\ t_1 := \frac{1}{t\_0}\\ t_2 := \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_1\right) \cdot u\\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{{t\_0}^{-2} - {t\_2}^{2}}{t\_1 - t\_2}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ (exp (/ (PI) s)) 1.0))
        (t_1 (/ 1.0 t_0))
        (t_2 (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_1) u)))
   (*
    (- s)
    (log (- (/ 1.0 (/ (- (pow t_0 -2.0) (pow t_2 2.0)) (- t_1 t_2))) 1.0)))))

Derivation

1. Initial program 99.0%

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

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

Alternative 2: 6.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{1}{1 + e^{t\_0}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_1\right) + t\_1} - 1\right) \leq -5.000000015855384 \cdot 10^{-29}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(t\_0, -0.25, 0.5\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right), -4, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ 1.0 (+ 1.0 (exp t_0)))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_1)) t_1))
           1.0)))
        -5.000000015855384e-29)
     (* (- s) (log (- (/ 1.0 (fma t_0 -0.25 0.5)) 1.0)))
     (fma (* (PI) (fma -0.5 u 0.25)) -4.0 0.0))))

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)))) < -5.00000002e-29

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 2.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{0.5 - \frac{\frac{\mathsf{fma}\left(-0.125, {u}^{3}, 0.015625\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(u \cdot u, 0.25, 0.0625 - u \cdot -0.125\right)}}{s}} - 1\right) \]

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 8.6%

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

   12. Applied rewrites 8.6%

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

   if -5.00000002e-29 < (*.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 98.9%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   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 15.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right), -4, 0\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 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)))))

Derivation

1. Initial program 99.0%

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

Alternative 4: 97.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in u around -inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. lower-/.f32 N/A

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

9. Applied rewrites 97.6%

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

Alternative 5: 85.7% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. associate-*r/ N/A

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

   4. unpow2 N/A

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

   5. times-frac N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f32 N/A

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

   17. lower-/.f32 N/A

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

   18. lower-PI.f32 95.7

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

5. Applied rewrites 95.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.7

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

7. Applied rewrites 39.0%

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

8. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 36.6

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

10. Applied rewrites 36.6%

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

11. Taylor expanded in s around inf

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

13. Applied rewrites 88.0%

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

Alternative 6: 6.0% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

5. Applied rewrites -0.0%

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

7. Applied rewrites 2.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{0.5 - \frac{\frac{\mathsf{fma}\left(-0.125, {u}^{3}, 0.015625\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(u \cdot u, 0.25, 0.0625 - u \cdot -0.125\right)}}{s}} - 1\right) \]

8. Taylor expanded in u around 0

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

10. Applied rewrites 10.3%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0, u, 0.5 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{u}, \mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right)\right)} - 1\right) \]
11. Step-by-step derivation

12. Applied rewrites 4.3%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0, u, 0.5 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right), u, 0.5 + -0.25 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]
13. Add Preprocessing

Alternative 7: 6.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ \mathbf{if}\;s \leq 7.999999837289519 \cdot 10^{-31}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(0.5 \cdot t\_0, u, \mathsf{fma}\left(-0.25, t\_0, 0.5\right)\right)} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(t\_0, -0.25, 0.5\right)} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)))
   (if (<= s 7.999999837289519e-31)
     (* (- s) (log (- (/ 1.0 (fma (* 0.5 t_0) u (fma -0.25 t_0 0.5))) 1.0)))
     (* (- s) (log (- (/ 1.0 (fma t_0 -0.25 0.5)) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if s < 7.99999984e-31

   1. Initial program 99.1%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 2.8%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{0.5 - \frac{\frac{\mathsf{fma}\left(-0.125, {u}^{3}, 0.015625\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(u \cdot u, 0.25, 0.0625 - u \cdot -0.125\right)}}{s}} - 1\right) \]

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 16.1%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0, u, 0.5 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{u}, \mathsf{fma}\left(-0.25, \frac{\mathsf{PI}\left(\right)}{s}, 0.5\right)\right)} - 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 15.7%

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

   if 7.99999984e-31 < s

   1. Initial program 99.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   5. Applied rewrites -0.0%

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

   7. Applied rewrites 2.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{0.5 - \frac{\frac{\mathsf{fma}\left(-0.125, {u}^{3}, 0.015625\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(u \cdot u, 0.25, 0.0625 - u \cdot -0.125\right)}}{s}} - 1\right) \]

   8. Taylor expanded in u around 0

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

   10. Applied rewrites 8.8%

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

   12. Applied rewrites 8.8%

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

Alternative 8: 21.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\left(\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(0.5 \cdot \mathsf{PI}\left(\right), u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 2\right) - -2 \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (-
     (fma -4.0 (/ (fma (* 0.5 (PI)) u (* 0.25 (PI))) s) 2.0)
     (* -2.0 (/ (PI) s)))
    1.0))))

Derivation

1. Initial program 99.0%

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

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

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 22.2%

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

Alternative 9: 11.6% accurate, 17.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
6. 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. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-PI.f32 11.1

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

7. Applied rewrites 11.1%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 11.3%

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

11. Final simplification 11.3%

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

Alternative 10: 11.6% accurate, 26.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
6. 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. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-PI.f32 11.1

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

7. Applied rewrites 11.1%

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

9. Applied rewrites 11.3%

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

10. Final simplification 11.3%

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

Alternative 11: 7.4% accurate, 30.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
6. 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. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-PI.f32 11.1

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

7. Applied rewrites 11.1%

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

8. Final simplification 11.1%

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

Alternative 12: 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 99.0%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 11.1

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

5. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024323 
(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))))