Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 25.7s

Alternatives: 12

Speedup: 1.0×

• Could not converge on a ground truth

  1. u = 6.950233455427224e-6
  2. s = 1.222770720232802e-7

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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\\ t_1 := \mathsf{fma}\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - t\_0, u, t\_0\right)\\ t_2 := \left({t\_1}^{-2} + 1\right) + \frac{1}{t\_1}\\ t_3 := \frac{1}{t\_2}\\ t_4 := \frac{{t\_1}^{-3}}{t\_2}\\ \left(-s\right) \cdot \log \left(\frac{{t\_4}^{3} - {t\_3}^{3}}{\mathsf{fma}\left(t\_4, t\_4, \mathsf{fma}\left(t\_3, t\_3, t\_4 \cdot t\_3\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (- (log1p (exp (/ (PI) s))))))
        (t_1 (fma (- (exp (- (log1p (exp (/ (- (PI)) s))))) t_0) u t_0))
        (t_2 (+ (+ (pow t_1 -2.0) 1.0) (/ 1.0 t_1)))
        (t_3 (/ 1.0 t_2))
        (t_4 (/ (pow t_1 -3.0) t_2)))
   (*
    (- s)
    (log
     (/
      (- (pow t_4 3.0) (pow t_3 3.0))
      (fma t_4 t_4 (fma t_3 t_3 (* t_4 t_3))))))))

Derivation

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

4. Applied rewrites 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\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)}^{-3}}{\left({\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\right) + \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)}}\right)}^{3} - {\left(\frac{1}{\left({\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\right) + \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)}}\right)}^{3}}{\mathsf{fma}\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)}^{-3}}{\left({\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\right) + \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)}}, \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)}^{-3}}{\left({\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\right) + \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)}}, \mathsf{fma}\left(\frac{1}{\left({\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\right) + \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)}}, \frac{1}{\left({\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\right) + \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)}}, \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)}^{-3}}{\left({\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\right) + \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)}} \cdot \frac{1}{\left({\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\right) + \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)}}\right)\right)}\right)} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\mathsf{log1p}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}\right)}\\ t_1 := \mathsf{fma}\left(e^{-\mathsf{log1p}\left(e^{\frac{-\mathsf{PI}\left(\right)}{s}}\right)} - t\_0, u, t\_0\right)\\ \left(-s\right) \cdot \log \left(\frac{{t\_1}^{-3} - 1}{\left({t\_1}^{-2} + 1\right) + \frac{1}{t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (- (log1p (exp (/ (PI) s))))))
        (t_1 (fma (- (exp (- (log1p (exp (/ (- (PI)) s))))) t_0) u t_0)))
   (*
    (- s)
    (log (/ (- (pow t_1 -3.0) 1.0) (+ (+ (pow t_1 -2.0) 1.0) (/ 1.0 t_1)))))))

Derivation

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

4. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\frac{1}{e^{\frac{-\mathsf{PI}\left(\right)}{s}} + 1} - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}, u, \frac{1}{e^{\frac{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{3}}{s}} + 1}\right)} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (fma
      (- (/ 1.0 (+ (exp (/ (- (PI)) s)) 1.0)) (/ 1.0 (+ (exp (/ (PI) s)) 1.0)))
      u
      (/ 1.0 (+ (exp (/ (pow (cbrt (PI)) 3.0) s)) 1.0))))
    1.0))))

Derivation

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

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]
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 99.0%

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

7. Applied rewrites 99.0%

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

Alternative 4: 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 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

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]
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 99.0%

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

Alternative 5: 97.4% 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 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 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 97.7%

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

Alternative 6: 37.6% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

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

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]
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 99.0%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 38.3%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(0.5 - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}, u, \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right)} - 1\right) \]
9. Add Preprocessing

Alternative 7: 36.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\left(0.5 - \frac{1}{e^{\frac{\mathsf{PI}\left(\right)}{s}} + 1}\right) \cdot u} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (- (/ 1.0 (* (- 0.5 (/ 1.0 (+ (exp (/ (PI) s)) 1.0))) u)) 1.0))))

Derivation

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 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 97.7%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 37.6%

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

Alternative 8: 24.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\mathsf{fma}\left(u \cdot \mathsf{PI}\left(\right), 0.5, -0.25 \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{s}, 1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma (fma (* u (PI)) 0.5 (* -0.25 (PI))) (/ -4.0 s) 1.0))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
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 25.6%

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

Alternative 9: 24.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot -0.5, 0.25 \cdot \mathsf{PI}\left(\right)\right)}{s}, 1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma 4.0 (/ (fma u (* (PI) -0.5) (* 0.25 (PI))) s) 1.0))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 15.1%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 15.1%

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

9. Taylor expanded in s around -inf

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

    1. associate-*r/ N/A

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

    2. +-commutative N/A

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

    3. associate-*r/ N/A

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

    4. lower-fma.f32 N/A

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

11. Applied rewrites 25.6%

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

Alternative 10: 11.8% accurate, 20.4× speedup

Math

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

FPCore

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

Derivation

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

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 11.7

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

5. Applied rewrites 11.7%

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

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. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.9

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

8. Applied rewrites 11.9%

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

9. Taylor expanded in u around inf

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

11. Applied rewrites 11.9%

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

Alternative 11: 11.8% accurate, 36.4× speedup

Math

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

FPCore

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

Derivation

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

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 11.7

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

5. Applied rewrites 11.7%

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

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. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.9

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

8. Applied rewrites 11.9%

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

9. Taylor expanded in u around 0

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

11. Applied rewrites 11.9%

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

Alternative 12: 11.5% 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 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\mathsf{PI}\left(\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]
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 11.7

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

5. Applied rewrites 11.7%

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

Reproduce

herbie shell --seed 2024350 
(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))))