VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 96.9%

Time: 17.1s

Alternatives: 8

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1478619236.6093044

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: 6.5% 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: 96.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (fma
   (PI)
   (/ (log (cosh (* f (/ (PI) -4.0)))) (* (PI) (PI)))
   (/ (log (sinh (* (/ (PI) 4.0) f))) (- (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 97.4%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

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

   4. lift-log.f64 N/A

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

   5. lower--.f64 N/A

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

6. Applied rewrites 97.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-/.f64 N/A

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

8. Applied rewrites 97.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. div-add N/A

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 2: 97.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (/ (PI) 4.0))))
   (* (/ (- (log (cosh t_0)) (log (sinh t_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 97.4%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. log-div N/A

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

   4. lift-log.f64 N/A

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

   5. lower--.f64 N/A

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

6. Applied rewrites 97.8%

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

Alternative 3: 97.0% 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 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 97.4%

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

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 97.4

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

7. Applied rewrites 97.4%

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

Alternative 4: 96.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 2\\ t_1 := 0.5 \cdot \mathsf{PI}\left(\right)\\ -\frac{\mathsf{fma}\left(4, \log \left(\frac{2}{t\_1}\right) - \log f, \left(\left(2 \cdot f\right) \cdot \left(\mathsf{fma}\left(\left(t\_0 \cdot 2\right) \cdot 0.005208333333333333, -2, t\_0 \cdot 0.0625\right) \cdot t\_1\right)\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (PI) 2.0)) (t_1 (* 0.5 (PI))))
   (-
    (/
     (fma
      4.0
      (- (log (/ 2.0 t_1)) (log f))
      (*
       (*
        (* 2.0 f)
        (* (fma (* (* t_0 2.0) 0.005208333333333333) -2.0 (* t_0 0.0625)) t_1))
       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 96.6%

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

Alternative 5: 96.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 2\\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(t\_0 \cdot 2\right) \cdot 0.005208333333333333, -2, t\_0 \cdot 0.0625\right) \cdot f, 1, \frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (PI) 2.0)))
   (*
    (/ -1.0 (/ (PI) 4.0))
    (log
     (fma
      (* (fma (* (* t_0 2.0) 0.005208333333333333) -2.0 (* t_0 0.0625)) f)
      1.0
      (/ (/ 2.0 (* 0.5 (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

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

5. Applied rewrites 96.2%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 2\right) \cdot 0.005208333333333333, -2, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 0.0625\right) \cdot f, 1, \frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)\right)} \]

6. Final simplification 96.2%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 2\right) \cdot 0.005208333333333333, -2, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 0.0625\right) \cdot f, 1, \frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)\right) \]
7. Add Preprocessing

Alternative 6: 95.8% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (* (/ (log (/ 2.0 (* (* 0.5 (* t_0 t_0)) f))) (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 97.4%

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

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. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 95.7

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

7. Applied rewrites 95.7%

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

9. Applied rewrites 95.7%

   \[\leadsto \frac{\log \left(\frac{2}{\left(0.5 \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot f}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
10. Add Preprocessing

Alternative 7: 95.8% accurate, 4.4× speedup

Math

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

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. 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 97.4%

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

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. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 95.7

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

7. Applied rewrites 95.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 95.7%

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

Alternative 8: 95.8% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (* (/ (log (/ 4.0 (* (PI) f))) (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 97.4%

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

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. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 95.7

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

7. Applied rewrites 95.7%

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

9. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2024353 
(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)))))))))