Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.0s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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. Add Preprocessing

Alternative 2: 27.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \mathsf{PI}\left(\right)\\ t_1 := \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\\ \mathbf{if}\;\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - t\_1\right) + t\_1} - 1\right) \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5, u, t\_0\right)}{s}, -8, 2\right) - \frac{\mathsf{PI}\left(\right) \cdot 12}{s} \cdot -0.25\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-s\right) \cdot \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}, -0.5, t\_0\right), \frac{4}{s}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* 0.25 (PI))) (t_1 (/ 1.0 (+ 1.0 (exp (/ (PI) s))))))
   (if (<=
        (*
         (- s)
         (log
          (-
           (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- (PI)) s)))) t_1)) t_1))
           1.0)))
        -9.999999717180685e-10)
     (*
      (- s)
      (log
       (-
        (-
         (fma (/ (fma (* (* 0.5 (PI)) 0.5) u t_0) s) -8.0 2.0)
         (* (/ (* (PI) 12.0) s) -0.25))
        1.0)))
     (*
      (- s)
      (log (fma (fma (* u (pow (sqrt (PI)) 2.0)) -0.5 t_0) (/ 4.0 s) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -9.99999972e-10

   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

   4. Applied rewrites 97.7%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-3} + {\left(\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u\right)}^{3}}{{\left(e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1\right)}^{-2} + \left({\left(\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u\right)}^{2} - \left(\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right) \cdot u\right) \cdot e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)}}} - 1\right) \]

   5. Taylor expanded in s around inf

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

      1. lower--.f32 N/A

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

   7. Applied rewrites 19.6%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5, u, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, -8, 2\right) - \frac{\mathsf{PI}\left(\right) \cdot 12}{s} \cdot -0.25\right)} - 1\right) \]

   if -9.99999972e-10 < (*.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

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

   5. Applied rewrites 11.4%

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

   7. Applied rewrites 28.3%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 98.8%

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

Alternative 4: 25.0% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 10.1%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 25.3%

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

Alternative 5: 11.9% accurate, 7.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-/.f32 10.9

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

5. Applied rewrites 10.8%

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

7. Applied rewrites 12.1%

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

Alternative 6: 12.2% accurate, 9.1× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (PI) u))) (* (fma (/ -1.0 u) t_0 (fma t_0 2.0 0.0)) (* u u))))

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 7.8%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 10.1%

   \[\leadsto 0 \]

9. Taylor expanded in u around inf

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

11. Applied rewrites 10.1%

    \[\leadsto \mathsf{fma}\left(\frac{-1}{u}, \frac{\mathsf{PI}\left(\right)}{u}, \left(0 + \frac{0}{u \cdot u}\right) + \mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{u}, 2, \frac{0}{u}\right)\right) \cdot \color{blue}{\left(u \cdot u\right)} \]

12. Final simplification 10.1%

    \[\leadsto \mathsf{fma}\left(\frac{-1}{u}, \frac{\mathsf{PI}\left(\right)}{u}, \mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{u}, 2, 0\right)\right) \cdot \left(u \cdot u\right) \]
13. Add Preprocessing

Alternative 7: 10.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0, -0.5, \frac{\mathsf{PI}\left(\right) \cdot \left(\left(u \cdot u\right) \cdot 0.25 - 0.0625\right)}{-0.5 \cdot u - 0.25} \cdot -4\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (fma
  0.0
  -0.5
  (* (/ (* (PI) (- (* (* u u) 0.25) 0.0625)) (- (* -0.5 u) 0.25)) -4.0)))

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 7.7%

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

7. Applied rewrites 10.8%

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

9. Applied rewrites 10.9%

   \[\leadsto \mathsf{fma}\left(0 \cdot \frac{-0.5}{s}, -0.5, \frac{\mathsf{PI}\left(\right) \cdot \left(\left(u \cdot u\right) \cdot 0.25 - 0.0625\right)}{-0.5 \cdot u - 0.25} \cdot -4\right) \]

10. Final simplification 11.0%

    \[\leadsto \mathsf{fma}\left(0, -0.5, \frac{\mathsf{PI}\left(\right) \cdot \left(\left(u \cdot u\right) \cdot 0.25 - 0.0625\right)}{-0.5 \cdot u - 0.25} \cdot -4\right) \]
11. Add Preprocessing

Alternative 8: 11.5% accurate, 17.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.1%

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

7. Applied rewrites 10.8%

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

9. Applied rewrites 11.0%

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

Alternative 9: 11.3% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

Alternative 10: 10.3% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.1%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 10.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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))))