VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.2%

Time: 17.7s

Alternatives: 7

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1445622456.3026845

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: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(\left(\left(f \cdot t\_0\right) \cdot t\_0\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (*
    (/
     (log (/ (cosh (* (/ (PI) -4.0) f)) (sinh (* (* (* f t_0) t_0) 0.25))))
     (PI))
    -4.0)))

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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 97.2

      \[\leadsto \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
8. Step-by-step derivation

9. Applied rewrites 97.2%

   \[\leadsto \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(\left(\left(f \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
10. Add Preprocessing

Alternative 2: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ (log (/ (cosh (* (/ (PI) -4.0) f)) (sinh (* (* (PI) f) 0.25)))) (PI))
  -4.0))

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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 97.2

      \[\leadsto \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
8. Add Preprocessing

Alternative 3: 96.7% accurate, 2.0× speedup

Math

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

FPCore

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

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

5. Applied rewrites 96.9%

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

Alternative 4: 96.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathsf{fma}\left(\frac{4}{t\_0} \cdot \log \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \mathsf{PI}\left(\right), \left(\left(t\_0 \cdot 0.08333333333333333\right) \cdot f\right) \cdot \frac{-f}{\mathsf{PI}\left(\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (fma
    (* (/ 4.0 t_0) (log (* (* f (PI)) 0.25)))
    (PI)
    (* (* (* t_0 0.08333333333333333) f) (/ (- f) (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

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left(2 \cdot f\right) \cdot -0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.041666666666666664, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f, -\mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \left(\left(-\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \cdot -4\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 96.9%

   \[\leadsto \mathsf{fma}\left(\log \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{-4}{\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}, \frac{\left(\left(\left(-1 \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333\right)\right) \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \]

11. Applied rewrites 96.9%

    \[\leadsto \mathsf{fma}\left(\frac{4}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)} \cdot \log \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \color{blue}{\mathsf{PI}\left(\right)}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.08333333333333333\right) \cdot \left(-f\right)\right) \cdot \frac{f}{\mathsf{PI}\left(\right)}\right) \]

12. Final simplification 96.9%

    \[\leadsto \mathsf{fma}\left(\frac{4}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)} \cdot \log \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \mathsf{PI}\left(\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.08333333333333333\right) \cdot f\right) \cdot \frac{-f}{\mathsf{PI}\left(\right)}\right) \]
13. Add Preprocessing

Alternative 5: 96.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), \frac{-4}{\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \left(\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.08333333333333333\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (* (log (* 0.25 (* f (PI)))) (PI))
  (/ -4.0 (* (- (PI)) (PI)))
  (* (* (* f f) (PI)) -0.08333333333333333)))

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

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\left(\left(2 \cdot f\right) \cdot -0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.041666666666666664, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f, -\mathsf{PI}\left(\right), \mathsf{PI}\left(\right) \cdot \left(\left(-\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right)\right) \cdot -4\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(-\mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 96.9%

   \[\leadsto \mathsf{fma}\left(\log \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{-4}{\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}, \frac{\left(\left(\left(-1 \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333\right)\right) \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \]

10. Taylor expanded in f around 0

    \[\leadsto \mathsf{fma}\left(\log \left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), \frac{-4}{\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{12} \cdot \left({f}^{2} \cdot \mathsf{PI}\left(\right)\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 96.9%

    \[\leadsto \mathsf{fma}\left(\log \left(0.25 \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \mathsf{PI}\left(\right), \frac{-4}{\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \left(\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.08333333333333333\right) \]
13. Add Preprocessing

Alternative 6: 96.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{4}{f \cdot \mathsf{PI}\left(\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log (/ 4.0 (* f (PI)))) (PI)) -4.0))

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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 96.2

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

7. Applied rewrites 96.2%

   \[\leadsto \frac{\log \color{blue}{\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}}{\mathsf{PI}\left(\right)} \cdot -4 \]
8. Step-by-step derivation

9. Applied rewrites 96.2%

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

Alternative 7: 4.2% accurate, 38.4× speedup

Math

\[\begin{array}{l} \\ \left(-0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(f \cdot f\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (* -0.08333333333333333 (PI)) (* f f)))

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

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

5. Applied rewrites 96.9%

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

6. Taylor expanded in f around inf

   \[\leadsto {f}^{2} \cdot \color{blue}{\left(-4 \cdot \frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - -1 \cdot \log \left(\frac{1}{f}\right)}{{f}^{2} \cdot \mathsf{PI}\left(\right)} + -1 \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 51.8%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\mathsf{PI}\left(\right)}\right) - \log f}{\left(f \cdot f\right) \cdot \mathsf{PI}\left(\right)}, -4, \mathsf{PI}\left(\right) \cdot -0.08333333333333333\right) \cdot \color{blue}{\left(f \cdot f\right)} \]

9. Taylor expanded in f around inf

   \[\leadsto \left(\frac{-1}{12} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(f \cdot f\right) \]
10. Step-by-step derivation

11. Applied rewrites 4.3%

    \[\leadsto \left(-0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(f \cdot f\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024346 
(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)))))))))