VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.0% → 98.9%

Time: 13.0s

Alternatives: 8

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1511515899.0466092

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)) (t_1 (/ (PI) -4.0)))
   (*
    (/ -1.0 t_0)
    (log
     (pow
      (pow (/ (cosh (* t_1 f)) (sinh (* f t_0))) (pow (exp f) t_0))
      (pow (exp f) t_1))))))

Derivation

1. Initial program 7.6%

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\frac{\mathsf{fma}\left(e^{\frac{f}{-4} \cdot \mathsf{PI}\left(\right)}, 2, 2 \cdot {\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{f}{4}\right)}\right)}{4}}{\sinh \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/
   (log
    (/
     (/
      (fma (exp (* (/ f -4.0) (PI))) 2.0 (* 2.0 (pow (exp (PI)) (/ f 4.0))))
      4.0)
     (sinh (* (* (PI) f) 0.25))))
   (PI))
  -4.0))

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

   \[\leadsto \frac{\log \left(\frac{\color{blue}{\frac{\mathsf{fma}\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{f}{-4}\right)}, 2, 2 \cdot {\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{f}{4}\right)}\right)}{4}}}{\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. lift-exp.f64 N/A

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

   4. rem-log-exp N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-/.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-exp.f64 N/A

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in f around 0

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

11. Applied rewrites 99.0%

    \[\leadsto \frac{\log \left(\frac{\frac{\mathsf{fma}\left(e^{\frac{f}{-4} \cdot \mathsf{PI}\left(\right)}, 2, 2 \cdot {\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{f}{4}\right)}\right)}{4}}{\sinh \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
12. Add Preprocessing

Alternative 3: 97.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in f around 0

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

10. Applied rewrites 99.0%

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

    1. lift-log.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. log-div N/A

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

    4. lower--.f64 N/A

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

    5. lower-log.f64 N/A

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

12. Applied rewrites 99.0%

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

Alternative 4: 97.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in f around 0

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

10. Applied rewrites 99.0%

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

Alternative 5: 96.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(\left(0.03125 \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\sinh \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/
   (log
    (/ (fma (* (* 0.03125 f) f) (* (PI) (PI)) 1.0) (sinh (* (* (PI) f) 0.25))))
   (PI))
  -4.0))

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{\color{blue}{1 + \frac{1}{32} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
6. Step-by-step derivation

7. Applied rewrites 98.0%

   \[\leadsto \frac{\log \left(\frac{\color{blue}{\mathsf{fma}\left(\left(0.03125 \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}}{\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

8. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{32} \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\sinh \color{blue}{\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
9. Step-by-step derivation

10. Applied rewrites 98.0%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(0.03125 \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\sinh \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
11. Add Preprocessing

Alternative 6: 96.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \left(0.03125 \cdot f\right) \cdot f, 1\right)}{\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f}\right) \cdot -4}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (*
   (log (/ (fma (* (PI) (PI)) (* (* 0.03125 f) f) 1.0) (* (* 0.25 (PI)) f)))
   -4.0)
  (PI)))

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{\color{blue}{1 + \frac{1}{32} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}}{\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
6. Step-by-step derivation

7. Applied rewrites 98.0%

   \[\leadsto \frac{\log \left(\frac{\color{blue}{\mathsf{fma}\left(\left(0.03125 \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}}{\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

8. Taylor expanded in f around 0

   \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{32} \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\color{blue}{\frac{1}{2} \cdot \left(f \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
9. Step-by-step derivation

10. Applied rewrites 97.6%

    \[\leadsto \frac{\log \left(\frac{\mathsf{fma}\left(\left(0.03125 \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\color{blue}{\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot 0.5}}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{32} \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot \frac{1}{2}}\right)}{\mathsf{PI}\left(\right)} \cdot -4} \]

    2. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{32} \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot \frac{1}{2}}\right)}{\mathsf{PI}\left(\right)}} \cdot -4 \]

    3. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{32} \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot \frac{1}{2}}\right) \cdot -4}{\mathsf{PI}\left(\right)}} \]

    4. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{32} \cdot f\right) \cdot f, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{\left(\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot \frac{1}{2}}\right) \cdot -4}{\mathsf{PI}\left(\right)}} \]

12. Applied rewrites 97.6%

    \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \left(0.03125 \cdot f\right) \cdot f, 1\right)}{\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f}\right) \cdot -4}{\mathsf{PI}\left(\right)}} \]
13. Add Preprocessing

Alternative 7: 96.1% accurate, 4.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

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

7. Applied rewrites 97.4%

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

9. Applied rewrites 97.5%

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

Alternative 8: 96.2% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.6%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 99.0%

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

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

7. Applied rewrites 97.4%

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

9. Applied rewrites 97.4%

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

Reproduce

herbie shell --seed 2025022 
(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)))))))))