VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 98.9%

Time: 17.0s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1387673767.9065266

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{4}\\ t_1 := t\_0 \cdot f\\ t_2 := e^{t\_1}\\ t_3 := e^{-t\_1}\\ -\frac{1}{t\_0} \cdot \log \left(\frac{t\_2 + t\_3}{t\_2 - t\_3}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)) (t_1 (* t_0 f)) (t_2 (exp t_1)) (t_3 (exp (- t_1))))
   (- (* (/ 1.0 t_0) (log (/ (+ t_2 t_3) (- t_2 t_3)))))))

Initial Program: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{4}\\ t_1 := t\_0 \cdot f\\ t_2 := e^{t\_1}\\ t_3 := e^{-t\_1}\\ -\frac{1}{t\_0} \cdot \log \left(\frac{t\_2 + t\_3}{t\_2 - t\_3}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)) (t_1 (* t_0 f)) (t_2 (exp t_1)) (t_3 (exp (- t_1))))
   (- (* (/ 1.0 t_0) (log (/ (+ t_2 t_3) (- t_2 t_3)))))))

Alternative 1: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\\ \left(\log \tanh \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot t\_0\right) \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt (PI)))))
   (* (* (log (tanh (* 0.25 (* f (PI))))) t_0) t_0)))

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

   6. sqr-pow N/A

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

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. unpow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{4}{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. sqrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt{4}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{2}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \frac{2}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-sqrt.f64 98.6

      \[\leadsto \frac{2}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot \log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\log \tanh \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)} \]

   12. lower-*.f64 98.6

       \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5}\right)} \]

6. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \left(\frac{f}{\frac{4}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\frac{f}{\frac{4}{\mathsf{PI}\left(\right)}}\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \left(\frac{f}{\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}}}\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   3. associate-/r/ N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\frac{f}{4} \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{f}{4}\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   5. associate-/l* N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot f}{4}\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   6. div-inv N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \color{blue}{\frac{1}{4}}\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot \frac{1}{4}\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \left(\color{blue}{\left(f \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{4}\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

   10. lower-*.f64 98.6

       \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \left(\color{blue}{\left(f \cdot \mathsf{PI}\left(\right)\right)} \cdot 0.25\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

8. Applied rewrites 98.6%

   \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \color{blue}{\left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)} \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

9. Final simplification 98.6%

   \[\leadsto \left(\log \tanh \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \mathsf{PI}\left(\right)\\ \frac{\log \tanh \left(t\_0 \cdot f\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI)))) (/ (log (tanh (* t_0 f))) t_0)))

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)}} \]

5. Final simplification 98.6%

   \[\leadsto \frac{\log \tanh \left(\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)}{0.25 \cdot \mathsf{PI}\left(\right)} \]
6. Add Preprocessing

Alternative 3: 96.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 2\\ \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_0 \cdot 2\right) \cdot 0.005208333333333333, -2, 0.0625 \cdot t\_0\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right) \cdot \frac{-4}{\mathsf{PI}\left(\right)} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (PI) 2.0)))
   (*
    (log
     (/
      (fma
       (* (fma (* (* t_0 2.0) 0.005208333333333333) -2.0 (* 0.0625 t_0)) f)
       f
       (/ 2.0 (* 0.5 (PI))))
      f))
    (/ -4.0 (PI)))))

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 94.8

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

5. Applied rewrites 94.8%

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

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

7. Applied rewrites 94.8%

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

8. Taylor expanded in f around 0

   \[\leadsto \frac{-4}{\mathsf{PI}\left(\right)} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

10. Applied rewrites 95.5%

    \[\leadsto \frac{-4}{\mathsf{PI}\left(\right)} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.005208333333333333 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 2\right), -2, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 0.0625\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right)} \]

11. Final simplification 95.5%

    \[\leadsto \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 2\right) \cdot 0.005208333333333333, -2, 0.0625 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right) \cdot \frac{-4}{\mathsf{PI}\left(\right)} \]
12. Add Preprocessing

Alternative 4: 96.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \left(\frac{-2}{t\_0} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \cdot \frac{2}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (* (* (/ (- 2.0) t_0) (log (/ (/ 4.0 (PI)) f))) (/ 2.0 t_0))))

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 94.8

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

5. Applied rewrites 94.8%

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

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{4}}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \]

7. Applied rewrites 94.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(4\right)}}{\mathsf{PI}\left(\right)} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{4}{\mathsf{PI}\left(\right)}\right)\right)} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{4}{\mathsf{PI}\left(\right)}}\right)\right) \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{4}{\mathsf{PI}\left(\right)} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)} \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{neg}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)\right)} \]

9. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(-\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right)\right)} \]

10. Final simplification 95.0%

    \[\leadsto \left(\frac{-2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \]
11. Add Preprocessing

Alternative 5: 96.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (log (* 0.25 (* f (PI)))) (* 0.25 (PI))))

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right)}{\frac{\mathsf{PI}\left(\right)}{4}}}\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)}} \]

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 95.0

      \[\leadsto \frac{\log \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot f\right) \cdot 0.25\right)}{0.25 \cdot \mathsf{PI}\left(\right)} \]

7. Applied rewrites 95.0%

   \[\leadsto \frac{\log \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}{0.25 \cdot \mathsf{PI}\left(\right)} \]

8. Final simplification 95.0%

   \[\leadsto \frac{\log \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}{0.25 \cdot \mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024256 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ (PI) 4.0)) (log (/ (+ (exp (* (/ (PI) 4.0) f)) (exp (- (* (/ (PI) 4.0) f)))) (- (exp (* (/ (PI) 4.0) f)) (exp (- (* (/ (PI) 4.0) f)))))))))