VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 96.7%

Time: 15.2s

Alternatives: 8

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1000142272.5446042

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: 7.1% 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: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{-4} \cdot f\\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \left(\log \left(\cosh t\_0 \cdot 2\right) - \log \left(-2 \cdot \sinh t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ (PI) -4.0) f)))
   (*
    (/ -1.0 (/ (PI) 4.0))
    (- (log (* (cosh t_0) 2.0)) (log (* -2.0 (sinh t_0)))))))

Derivation

1. Initial program 7.2%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

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

   4. lower--.f64 N/A

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

4. Applied rewrites 97.4%

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

5. Final simplification 97.4%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \left(\log \left(\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right) \cdot 2\right) - \log \left(-2 \cdot \sinh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)\right)\right) \]
6. Add Preprocessing

Alternative 2: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

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

4. Applied rewrites 97.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r/ N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r/ N/A

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

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

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 97.3%

   \[\leadsto \color{blue}{-4 \cdot \left(\frac{\log \left(\cosh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{-4}\right) \cdot 2\right)}{\mathsf{PI}\left(\right)} - \frac{\log \left(\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{-4}\right) \cdot -2\right)}{\mathsf{PI}\left(\right)}\right)} \]
7. Add Preprocessing

Alternative 3: 96.9% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 97.3%

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

Alternative 4: 96.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -\frac{\mathsf{fma}\left(4, \log \left(\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right) - \log f, \left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (fma
    4.0
    (- (log (/ 2.0 (* 0.5 (PI)))) (log f))
    (* (* (* (* (PI) 0.08333333333333333) (PI)) f) f))
   (PI))))

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in f around 0

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

8. Applied rewrites 97.2%

   \[\leadsto -\frac{\mathsf{fma}\left(4, \log \left(\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right) - \log f, \left(\left(\left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Alternative 5: 96.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ -\frac{\mathsf{fma}\left(\log \left(\frac{\frac{4}{\mathsf{PI}\left(\right)}}{f}\right), 4, \left(f \cdot \left(2 \cdot f\right)\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.041666666666666664, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5\right)\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (/
   (fma
    (log (/ (/ 4.0 (PI)) f))
    4.0
    (*
     (* f (* 2.0 f))
     (* (* (fma (PI) -0.041666666666666664 (* 0.125 (PI))) (PI)) 0.5)))
   (PI))))

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in f around 0

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 96.7%

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

Alternative 6: 96.0% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 96.0

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{2}{0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{f}\right) \]

5. Applied rewrites 96.0%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 96.0%

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

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 96.6%

    \[\leadsto \frac{-4}{\mathsf{PI}\left(\right)} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{3} \cdot \frac{0.005208333333333333}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25}, -2, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot 0.125\right), f, 0\right), f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right)} \]

11. Taylor expanded in f around 0

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

13. Applied rewrites 96.6%

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

Alternative 7: 95.5% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 96.0

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{2}{0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{f}\right) \]

5. Applied rewrites 96.0%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 96.0%

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

9. Applied rewrites 96.0%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

11. Applied rewrites 96.1%

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

Alternative 8: 95.4% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.2%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 96.0

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\frac{2}{0.5 \cdot \color{blue}{\mathsf{PI}\left(\right)}}}{f}\right) \]

5. Applied rewrites 96.0%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 96.0%

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

9. Applied rewrites 96.0%

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

Reproduce

herbie shell --seed 2025006 
(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)))))))))