Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 22.9s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

   1. lift-/.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. lower-/.f32 N/A

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

   4. lower-/.f32 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 16.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{-1}{e^{t\_0} - -1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;\log \left(\left(t\_0 + 1\right) - \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{-0.5}{s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ -1.0 (- (exp t_0) -1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/
            -1.0
            (- (* (- t_1 (/ -1.0 (- (exp (/ (- (PI)) s)) -1.0))) u) t_1))))
         (- s))
        -9.999999682655225e-20)
     (* (log (- (+ t_0 1.0) (* (/ (* (PI) (PI)) s) (/ -0.5 s)))) (- s))
     (* (log (fma (/ (fma -0.5 (* (PI) u) (* 0.25 (PI))) s) 4.0 1.0)) (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -9.99999968e-20

   1. Initial program 99.3%

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

   3. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in u around 0

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

   8. Applied rewrites 25.4%

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

   if -9.99999968e-20 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 10.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 4.5%

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

   9. Taylor expanded in s around -inf

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

       1. associate-*r/ N/A

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

       2. +-commutative N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

   11. Applied rewrites 9.5%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}}\right) \cdot \left(-s\right) \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;\log \left(\left(\frac{\mathsf{PI}\left(\right)}{s} + 1\right) - \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s} \cdot \frac{-0.5}{s}\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 15.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_0 - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - t\_0}\right) \cdot \left(-s\right) \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\log \left(\left(\frac{u \cdot u}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot 2\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ -1.0 (- (exp (/ (PI) s)) -1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/
            -1.0
            (- (* (- t_0 (/ -1.0 (- (exp (/ (- (PI)) s)) -1.0))) u) t_0))))
         (- s))
        -9.999999682655225e-21)
     (* (log (* (* (/ (* u u) s) (/ (* (PI) (PI)) s)) 2.0)) (- s))
     (* (log (fma (/ (fma -0.5 (* (PI) u) (* 0.25 (PI))) s) 4.0 1.0)) (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -9.99999968e-21

   1. Initial program 99.3%

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

   3. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 9.6%

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

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 10.1%

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

   12. Taylor expanded in u around inf

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

   14. Applied rewrites 20.6%

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

   if -9.99999968e-21 < (*.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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 9.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 4.3%

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

   9. Taylor expanded in s around -inf

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

       1. associate-*r/ N/A

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

       2. +-commutative N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

   11. Applied rewrites 13.4%

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

4. Final simplification 14.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}}\right) \cdot \left(-s\right) \leq -9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\log \left(\left(\frac{u \cdot u}{s} \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{s}\right) \cdot 2\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 14.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{s}\\ t_1 := \frac{-1}{e^{t\_0} - -1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -2.1999999560335437 \cdot 10^{-18}:\\ \;\;\;\;\log \left(\left(t\_0 \cdot t\_0\right) \cdot \left(\left(u \cdot u\right) \cdot 2\right)\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) s)) (t_1 (/ -1.0 (- (exp t_0) -1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/
            -1.0
            (- (* (- t_1 (/ -1.0 (- (exp (/ (- (PI)) s)) -1.0))) u) t_1))))
         (- s))
        -2.1999999560335437e-18)
     (* (log (* (* t_0 t_0) (* (* u u) 2.0))) (- s))
     (* (log (fma (/ (fma -0.5 (* (PI) u) (* 0.25 (PI))) s) 4.0 1.0)) (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -2.19999996e-18

   1. Initial program 99.3%

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

   3. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 20.2%

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

   if -2.19999996e-18 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.0%

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

   3. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 9.5%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 6.3%

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

   9. Taylor expanded in s around -inf

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

       1. associate-*r/ N/A

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

       2. +-commutative N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

   11. Applied rewrites 13.1%

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

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}}\right) \cdot \left(-s\right) \leq -2.1999999560335437 \cdot 10^{-18}:\\ \;\;\;\;\log \left(\left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right) \cdot \left(\left(u \cdot u\right) \cdot 2\right)\right) \cdot \left(-s\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 12.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ t_1 := \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{t\_0}{s}} - -1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -1.7999999428779406 \cdot 10^{-21}:\\ \;\;\;\;\log \mathsf{E}\left(\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))) (t_1 (/ -1.0 (- (exp (/ (PI) s)) -1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/ -1.0 (- (* (- t_1 (/ -1.0 (- (exp (/ t_0 s)) -1.0))) u) t_1))))
         (- s))
        -1.7999999428779406e-21)
     (* (log (E)) t_0)
     (* (log (fma (/ (fma -0.5 (* (PI) u) (* 0.25 (PI))) s) 4.0 1.0)) (- s)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.7999999e-21

   1. Initial program 99.3%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 15.2

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

   5. Applied rewrites 15.2%

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

   7. Applied rewrites 15.2%

      \[\leadsto -\mathsf{PI}\left(\right) \cdot \log \mathsf{E}\left(\right) \]

   if -1.7999999e-21 < (*.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

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   5. Applied rewrites 7.1%

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

   6. Taylor expanded in u around inf

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

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in s around -inf

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

       1. associate-*r/ N/A

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

       2. +-commutative N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

   11. Applied rewrites 13.4%

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

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}}\right) \cdot \left(-s\right) \leq -1.7999999428779406 \cdot 10^{-21}:\\ \;\;\;\;\log \mathsf{E}\left(\right) \cdot \left(-\mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right) \cdot u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 4, 1\right)\right) \cdot \left(-s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 14.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ t_1 := \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}\\ \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(t\_1 - \frac{-1}{e^{\frac{t\_0}{s}} - -1}\right) \cdot u - t\_1}\right) \cdot \left(-s\right) \leq -1.5000000170217692 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0 \cdot s\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))) (t_1 (/ -1.0 (- (exp (/ (PI) s)) -1.0))))
   (if (<=
        (*
         (log
          (-
           -1.0
           (/ -1.0 (- (* (- t_1 (/ -1.0 (- (exp (/ t_0 s)) -1.0))) u) t_1))))
         (- s))
        -1.5000000170217692e-19)
     t_0
     (* 0.0 s))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.50000002e-19

   1. Initial program 99.3%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 15.6

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

   5. Applied rewrites 15.6%

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

   if -1.50000002e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 99.0%

      \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
   2. Add Preprocessing
   3. 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. lower-/.f32 N/A

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

      4. lower-/.f32 99.0

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in s around inf

      \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(4 \cdot e^{\mathsf{neg}\left(\log 2\right)} - 1\right)\right)} \]
   8. 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 13.2

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

   9. Applied rewrites 13.2%

      \[\leadsto \color{blue}{0 \cdot s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(-1 - \frac{-1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} - -1}\right) \cdot u - \frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1}}\right) \cdot \left(-s\right) \leq -1.5000000170217692 \cdot 10^{-19}:\\ \;\;\;\;-\mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;0 \cdot s\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 8: 54.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}}{-s}} - -1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     1.0
     (*
      (-
       (/ -1.0 (- (exp (/ (PI) s)) -1.0))
       (/
        -1.0
        (-
         (exp
          (/ (/ (* (* (PI) (PI)) (PI)) (fma (PI) (PI) (* 0.0 (PI)))) (- s)))
         -1.0)))
      u))
    1.0))
  (- s)))

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 97.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\frac{0 - {\mathsf{PI}\left(\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \]
8. Step-by-step derivation

9. Applied rewrites 64.7%

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

10. Final simplification 96.9%

    \[\leadsto \log \left(\frac{1}{\left(\frac{-1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} - -1} - \frac{-1}{e^{\frac{\frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}}{-s}} - -1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \]
11. Add Preprocessing

Alternative 9: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 10: 8.2% accurate, 1.8× speedup

Math

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

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

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

5. Applied rewrites 8.0%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 9.2%

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

9. Taylor expanded in s around -inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -1 \cdot \frac{\left(-8 \cdot \frac{{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s} + -4 \cdot \frac{\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right) - 4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)} \]
10. 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

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

11. Applied rewrites 13.8%

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

12. Final simplification 12.9%

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

Alternative 11: 25.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* 0.5 (PI))))
   (*
    (log (- (- 2.0 (/ (* (- (+ (* t_0 u) (* t_0 0.5)) t_0) 4.0) s)) 1.0))
    (- s))))

Derivation

1. Initial program 99.1%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
2. Add Preprocessing
3. 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. lower-/.f32 N/A

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

   4. lower-/.f32 99.1

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

4. Applied rewrites 99.1%

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

6. Applied rewrites 97.3%

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

7. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

9. Applied rewrites 24.8%

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

10. Final simplification 24.8%

    \[\leadsto \log \left(\left(2 - \frac{\left(\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot u + \left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5\right) - 0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot 4}{s}\right) - 1\right) \cdot \left(-s\right) \]
11. Add Preprocessing

Alternative 12: 14.3% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.199999961918627 \cdot 10^{-20}:\\ \;\;\;\;0 \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2 - \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 1.199999961918627e-20) (* 0.0 s) (- (* (* (PI) u) 2.0) (PI))))

Derivation

1. Split input into 2 regimes

2. if s < 1.2e-20

   1. Initial program 99.1%

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

      1. lift-/.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. lower-/.f32 N/A

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

      4. lower-/.f32 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in s around inf

      \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(4 \cdot e^{\mathsf{neg}\left(\log 2\right)} - 1\right)\right)} \]
   8. 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 13.5

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

   9. Applied rewrites 13.5%

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

   if 1.2e-20 < s

   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 15.0

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

   5. Applied rewrites 15.0%

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

   6. Taylor expanded in s around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. lower-PI.f32 15.0

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

   8. Applied rewrites 15.0%

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

   10. Applied rewrites 15.4%

       \[\leadsto \left(u \cdot \mathsf{PI}\left(\right)\right) \cdot 2 - \color{blue}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.199999961918627 \cdot 10^{-20}:\\ \;\;\;\;0 \cdot s\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2 - \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 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 99.1%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 11.0

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

5. Applied rewrites 11.0%

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

Reproduce

herbie shell --seed 2024267 
(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))))