VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 99.0%

Time: 14.2s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1402920465.5275822

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: 6.8% 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: 99.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.9%

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

4. Applied rewrites 77.4%

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

   1. lift-log.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-to-exp N/A

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

   4. rem-log-exp N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 96.9%

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

8. Applied rewrites 99.3%

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

Alternative 2: 98.8% accurate, 2.6× speedup

Math

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

FPCore

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

4. Applied rewrites 77.4%

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

   1. lift-log.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-to-exp N/A

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

   4. rem-log-exp N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 96.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.8

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

   7. lift-/.f64 N/A

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

8. Applied rewrites 99.2%

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

Alternative 3: 96.0% accurate, 2.7× speedup

Math

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

FPCore

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

4. Applied rewrites 96.5%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. neg-log N/A

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

   4. lower-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 96.0%

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

11. Applied rewrites 96.0%

    \[\leadsto \frac{\log \left(\mathsf{PI}\left(\right) \cdot 0.25\right) + \log f}{\mathsf{PI}\left(\right)} \cdot 4 \]
12. Add Preprocessing

Alternative 4: 96.0% accurate, 4.8× speedup

Math

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

FPCore

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

4. Applied rewrites 96.5%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. neg-log N/A

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

   4. lower-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 96.0%

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

11. Applied rewrites 95.9%

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

Alternative 5: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.9%

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

4. Applied rewrites 96.5%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. neg-log N/A

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

   4. lower-log.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 96.0%

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

11. Applied rewrites 95.8%

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

Reproduce

herbie shell --seed 2024351 
(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)))))))))