Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 17.6s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 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, 0.8× speedup

Math

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

FPCore

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

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

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-PI.f32 N/A

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

   2. add-log-exp N/A

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

   3. *-un-lft-identity N/A

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

   4. lift-PI.f32 N/A

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

   5. exp-prod N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 99.1

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

6. Applied rewrites 99.1%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

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

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.1

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

4. Applied rewrites 99.1%

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

Alternative 3: 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)))))

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. Add Preprocessing

Alternative 4: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

5. Applied rewrites 97.6%

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

Alternative 5: 6.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, t\_0\right)\\ t_2 := {t\_1}^{2}\\ \mathbf{if}\;s \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_2, -16, \frac{1}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4\right)}^{-2}}\right)}{s}, -0.5, t\_1 \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_2, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), u \cdot -0.5, t\_0\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(t\_0, -4, \left(\left(-0.5 \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* 0.25 (PI)))
        (t_1 (fma (* u (PI)) -0.5 t_0))
        (t_2 (pow t_1 2.0)))
   (if (<= s 1.000000031374395e-22)
     (fma
      (/
       (fma t_2 -16.0 (/ 1.0 (pow (* (* (PI) (fma -0.5 u 0.25)) -4.0) -2.0)))
       s)
      -0.5
      (* t_1 -4.0))
     (fma
      (/ (fma t_2 -16.0 (fma 16.0 (pow (fma (PI) (* u -0.5) t_0) 2.0) 0.0)) s)
      -0.5
      (fma t_0 -4.0 (* (* (* -0.5 u) (PI)) -4.0))))))

Derivation

1. Split input into 2 regimes

2. if s < 1.00000003e-22

   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

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

   5. Applied rewrites 7.3%

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

   7. Applied rewrites 8.4%

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

   9. Applied rewrites 10.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \frac{1}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4\right)}^{-2}}\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

   if 1.00000003e-22 < 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 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 8.0%

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

   7. Applied rewrites 7.7%

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

   9. Applied rewrites 3.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), u \cdot -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \mathsf{fma}\left(0.25 \cdot \mathsf{PI}\left(\right), -4, \left(\left(-0.5 \cdot u\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 11.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;s \leq 9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_0}^{2}, -16, \frac{1}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4\right)}^{-2}}\right)}{s}, -0.5, t\_0 \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (fma (* u (PI)) -0.5 (* 0.25 (PI)))))
   (if (<= s 9.99999983775159e-18)
     (fma
      (/
       (fma
        (pow t_0 2.0)
        -16.0
        (/ 1.0 (pow (* (* (PI) (fma -0.5 u 0.25)) -4.0) -2.0)))
       s)
      -0.5
      (* t_0 -4.0))
     (* (- (* 2.0 (PI)) (/ (PI) u)) u))))

Derivation

1. Split input into 2 regimes

2. if s < 9.99999984e-18

   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

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 8.6%

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

   9. Applied rewrites 9.5%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \frac{1}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right) \cdot -4\right)}^{-2}}\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]

   if 9.99999984e-18 < 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 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 inf

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 15.9%

       \[\leadsto \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 11.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \mathsf{PI}\left(\right)\\ \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, t\_0\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {t\_0}^{2}, 0\right)\right)}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* 0.25 (PI))))
   (fma
    (/
     (fma
      (pow (fma (* u (PI)) -0.5 t_0) 2.0)
      -16.0
      (fma 16.0 (pow t_0 2.0) 0.0))
     s)
    -0.5
    (* (* 2.0 (PI)) u))))

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

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

5. Applied rewrites 8.2%

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

6. Taylor expanded in u around 0

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{-1}{2}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}, 0\right)\right)}{s}, \frac{-1}{2}, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{-1}{2}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]
7. Step-by-step derivation

8. Applied rewrites 11.8%

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

9. Taylor expanded in u around inf

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{-1}{2}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}, 0\right)\right)}{s}, \frac{-1}{2}, 2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 12.7%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(0.25 \cdot \mathsf{PI}\left(\right)\right)}^{2}, 0\right)\right)}{s}, -0.5, \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u\right) \]
12. Add Preprocessing

Alternative 8: 11.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;s \leq 4.0999998698886424 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)\right)}^{2}, -16, \mathsf{fma}\left(u \cdot t\_0, \mathsf{fma}\left(4, u, -4\right), t\_0\right)\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* (PI) (PI))))
   (if (<= s 4.0999998698886424e-21)
     (fma
      (/
       (fma
        (pow (* (PI) (fma -0.5 u 0.25)) 2.0)
        -16.0
        (fma (* u t_0) (fma 4.0 u -4.0) t_0))
       s)
      -0.5
      (* (fma (* u (PI)) -0.5 (* 0.25 (PI))) -4.0))
     (* (- (* 2.0 (PI)) (/ (PI) u)) u))))

Derivation

1. Split input into 2 regimes

2. if s < 4.09999987e-21

   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

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in u around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{-1}{2}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \mathsf{fma}\left(16, {\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}, 0\right)\right)}{s}, \frac{-1}{2}, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), \frac{-1}{2}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in u around 0

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

   11. Applied rewrites 8.6%

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

   13. Applied rewrites 8.0%

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

   if 4.09999987e-21 < 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 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 8.8%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 14.7%

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

   9. Taylor expanded in u around inf

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

   11. Applied rewrites 14.9%

       \[\leadsto \left(2 \cdot \mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{u}\right) \cdot u \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 11.6% accurate, 20.4× speedup

Math

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

FPCore

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

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

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

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 inf

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

8. Applied rewrites 11.3%

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

9. Taylor expanded in u around inf

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

11. Applied rewrites 11.4%

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

Alternative 10: 11.6% accurate, 36.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf

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

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 inf

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

8. Applied rewrites 11.3%

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

10. Applied rewrites 11.4%

    \[\leadsto \left(\mathsf{PI}\left(\right) \cdot u\right) \cdot 2 - \mathsf{PI}\left(\right) \]
11. Add Preprocessing

Alternative 11: 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 11.3

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

5. Applied rewrites 11.3%

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

Alternative 12: 10.3% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

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 8.3%

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

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 10.6%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))))