Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 13.9s

Alternatives: 6

Speedup: 0.3×

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: 99.0% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\mathsf{PI}\left(\right)}{s}}\\ t_1 := e^{-\mathsf{log1p}\left(t\_0\right)}\\ t_2 := e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)}\\ \left(-s\right) \cdot \log \left(\frac{{\left(\mathsf{fma}\left(t\_2 - \frac{1}{1 + t\_0}, u, t\_1\right)\right)}^{-2} - 1}{\frac{1}{\mathsf{fma}\left(t\_2 - t\_1, u, t\_1\right)} + 1}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ (PI) s)))
        (t_1 (exp (- (log1p t_0))))
        (t_2 (exp (- (log1p (exp (/ (- (PI)) s)))))))
   (*
    (- s)
    (log
     (/
      (- (pow (fma (- t_2 (/ 1.0 (+ 1.0 t_0))) u t_1) -2.0) 1.0)
      (+ (/ 1.0 (fma (- t_2 t_1) u t_1)) 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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. flip-- N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\mathsf{fma}\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}, u, e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)\right)}^{-2} - 1}{\frac{1}{\mathsf{fma}\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}, u, e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\right)} + 1}\right)} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. exp-neg N/A

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

   4. lift-log1p.f32 N/A

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

   5. rem-exp-log N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 99.0

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

6. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (fma (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u 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. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 98.9%

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

Alternative 3: 97.8% 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 97.8%

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

Alternative 4: 25.2% 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. +-commutative N/A

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

   2. associate-*r/ N/A

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

   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 24.3%

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

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 24.7%

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

Alternative 5: 11.6% accurate, 23.2× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (* (fma (* 0.5 (PI)) u (* -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 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 11.3

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

5. Applied rewrites 11.3%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

   4. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 11.6

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

8. Applied rewrites 11.6%

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

Alternative 6: 11.4% accurate, 170.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 11.3

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

5. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2024352 
(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))))