Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 17.5s

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: 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: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \log \left(\frac{1}{\frac{1}{e^{\frac{t\_0}{s} \cdot {t\_0}^{2}} + 1} + \left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \cdot \left(-s\right) \end{array} \end{array} \]

FPCore

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 4.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ t_1 := \mathsf{PI}\left(\right) \cdot u\\ t_2 := -0.25 \cdot \mathsf{PI}\left(\right)\\ t_3 := \mathsf{fma}\left(t\_1, 0.5, t\_2\right)\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_0\right) \cdot u + t\_0} - 1\right) \cdot \left(-s\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\frac{-{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \mathsf{PI}\left(\right), 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_3}^{2}, -16, \frac{1}{{\left(\mathsf{fma}\left(0.5, t\_1, t\_2\right) \cdot 4\right)}^{-2}}\right)}{s}, -0.5, 4 \cdot t\_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
        (t_1 (* (PI) u))
        (t_2 (* -0.25 (PI)))
        (t_3 (fma t_1 0.5 t_2)))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u) t_0))
           1.0))
         (- s))
        -1.999999936531045e-19)
     (/ (- (pow (PI) 3.0)) (fma (PI) (fma 1.0 (PI) 0.0) (* 0.0 (PI))))
     (fma
      (/
       (fma (pow t_3 2.0) -16.0 (/ 1.0 (pow (* (fma 0.5 t_1 t_2) 4.0) -2.0)))
       s)
      -0.5
      (* 4.0 t_3)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32)))) < -1.99999994e-19

   1. Initial program 99.0%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 14.4

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

   5. Applied rewrites 14.4%

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

   7. Applied rewrites 8.1%

      \[\leadsto \frac{0 - {\mathsf{PI}\left(\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.0%

      \[\leadsto \frac{0 - {\mathsf{PI}\left(\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \color{blue}{\mathsf{PI}\left(\right)}, 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)} \]

   if -1.99999994e-19 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 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

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 7.6%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \frac{1}{{\left(4 \cdot \mathsf{fma}\left(0.5, u \cdot \mathsf{PI}\left(\right), -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{-2}}\right)}{s}, -0.5, \mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\frac{-{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \mathsf{PI}\left(\right), 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \frac{1}{{\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right) \cdot u, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right)}^{-2}}\right)}{s}, -0.5, 4 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 5.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ t_1 := \mathsf{PI}\left(\right) \cdot u\\ t_2 := -0.25 \cdot \mathsf{PI}\left(\right)\\ t_3 := \mathsf{fma}\left(0.5, t\_1, t\_2\right)\\ t_4 := \mathsf{fma}\left(t\_1, 0.5, t\_2\right)\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - t\_0\right) \cdot u + t\_0} - 1\right) \cdot \left(-s\right) \leq -6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\frac{-{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \mathsf{PI}\left(\right), 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left({t\_4}^{2}, -16, \left(\left(t\_3 \cdot 4\right) \cdot 4\right) \cdot t\_3\right)}{s}, -0.5, 4 \cdot t\_4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
        (t_1 (* (PI) u))
        (t_2 (* -0.25 (PI)))
        (t_3 (fma 0.5 t_1 t_2))
        (t_4 (fma t_1 0.5 t_2)))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u) t_0))
           1.0))
         (- s))
        -6.000000068087077e-18)
     (/ (- (pow (PI) 3.0)) (fma (PI) (fma 1.0 (PI) 0.0) (* 0.0 (PI))))
     (fma
      (/ (fma (pow t_4 2.0) -16.0 (* (* (* t_3 4.0) 4.0) t_3)) s)
      -0.5
      (* 4.0 t_4)))))

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)))) < -6.00000007e-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 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 14.6

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

   5. Applied rewrites 14.6%

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

   7. Applied rewrites 8.2%

      \[\leadsto \frac{0 - {\mathsf{PI}\left(\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.1%

      \[\leadsto \frac{0 - {\mathsf{PI}\left(\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \color{blue}{\mathsf{PI}\left(\right)}, 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)} \]

   if -6.00000007e-18 < (*.f32 (neg.f32 s) (log.f32 (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 (*.f32 u (-.f32 (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (PI.f32) s)))))) #s(literal 1 binary32))))

   1. Initial program 98.8%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 6.2%

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

   7. Applied rewrites 6.4%

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

4. Final simplification 4.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -6.000000068087077 \cdot 10^{-18}:\\ \;\;\;\;\frac{-{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \mathsf{PI}\left(\right), 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}, -16, \left(\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right) \cdot u, -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot 4\right) \cdot 4\right) \cdot \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right) \cdot u, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}{s}, -0.5, 4 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u, 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 14.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{PI}\left(\right)\\ t_1 := \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\\ \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{t\_0}{s}} + 1} - t\_1\right) \cdot u + t\_1} - 1\right) \cdot \left(-s\right) \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (PI))) (t_1 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (if (<=
        (*
         (log
          (-
           (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ t_0 s)) 1.0)) t_1) u) t_1))
           1.0))
         (- s))
        -9.999999682655225e-20)
     t_0
     0.0)))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.0%

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

   3. Taylor expanded in u around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-PI.f32 14.3

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

   5. Applied rewrites 14.3%

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

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

   1. Initial program 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

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 13.6%

      \[\leadsto 0 \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{1}{\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u + \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}} - 1\right) \cdot \left(-s\right) \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;-\mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
   (*
    (log
     (-
      (/ 1.0 (+ (* (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) t_0) u) t_0))
      1.0))
    (- 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. Final simplification 98.9%

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

Alternative 6: 97.5% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log
   (-
    (/
     (/ 1.0 u)
     (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) (/ 1.0 (+ (exp (/ (PI) s)) 1.0))))
    1.0))
  (- 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 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 98.1%

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

6. Final simplification 98.1%

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

Alternative 7: 0.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{-{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \mathsf{PI}\left(\right), 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (/ (- (pow (PI) 3.0)) (fma (PI) (fma 1.0 (PI) 0.0) (* 0.0 (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 10.8

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

5. Applied rewrites 10.8%

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

7. Applied rewrites 6.7%

   \[\leadsto \frac{0 - {\mathsf{PI}\left(\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 0 \cdot \mathsf{PI}\left(\right)\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 0.0%

   \[\leadsto \frac{0 - {\mathsf{PI}\left(\right)}^{3}}{0 + \mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \color{blue}{\mathsf{PI}\left(\right)}, 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)} \]

10. Final simplification 0.1%

    \[\leadsto \frac{-{\mathsf{PI}\left(\right)}^{3}}{\mathsf{fma}\left(\mathsf{PI}\left(\right), \mathsf{fma}\left(1, \mathsf{PI}\left(\right), 0\right), 0 \cdot \mathsf{PI}\left(\right)\right)} \]
11. Add Preprocessing

Alternative 8: 11.7% 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 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

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

5. Applied rewrites 7.6%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 10.8%

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

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 11.0%

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

Alternative 9: 11.7% 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 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

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

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

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

8. Applied rewrites 10.8%

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

10. Applied rewrites 11.0%

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

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

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. Taylor expanded in s around 0

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

8. Applied rewrites 10.7%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(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))))