VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.1%

Time: 16.1s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1042584100.4010968

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: 6.9% 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: 99.1% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

   6. sqr-pow N/A

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow1/2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow1/2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. frac-times N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. times-frac N/A

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

   16. frac-2neg N/A

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

   17. metadata-eval N/A

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

   18. lift-neg.f64 N/A

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

   19. times-frac N/A

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

8. Applied rewrites 99.0%

   \[\leadsto \frac{4 \cdot \log \tanh \color{blue}{\left(\frac{-1}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \frac{f}{\frac{-2}{\sqrt{\mathsf{PI}\left(\right)}}}\right)}}{\mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\\ t\_0 \cdot \left(\log \tanh \left(\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt (PI)))))
   (* t_0 (* (log (tanh (* (* 0.25 (PI)) f))) t_0))))

Derivation

1. Initial program 7.3%

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

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

   6. sqr-pow N/A

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lower-/.f64 99.0

      \[\leadsto \color{blue}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.0

       \[\leadsto \frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5}\right)} \]

6. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\log \tanh \left(\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]
7. Add Preprocessing

Alternative 3: 99.1% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI)))) (/ (log (tanh (* f t_0))) t_0)))

Derivation

1. Initial program 7.3%

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

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

4. Applied rewrites 99.0%

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

Alternative 4: 96.2% accurate, 3.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 96.1%

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

6. Final simplification 96.1%

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

Alternative 5: 95.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{\mathsf{PI}\left(\right)}}\\ t\_0 \cdot \left(t\_0 \cdot \log \left(\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt (PI))))) (* t_0 (* t_0 (log (* (* 0.25 (PI)) f))))))

Derivation

1. Initial program 7.3%

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

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

5. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-log.f64 95.4

      \[\leadsto \frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\log \left(0.25 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\log f}}} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\log \left(0.25 \cdot \mathsf{PI}\left(\right)\right) + \log f}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

9. Applied rewrites 95.6%

   \[\leadsto \color{blue}{\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \log \left(\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right)} \]
10. Add Preprocessing

Alternative 6: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

5. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-log.f64 95.4

      \[\leadsto \frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\log \left(0.25 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\log f}}} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\log \left(0.25 \cdot \mathsf{PI}\left(\right)\right) + \log f}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

9. Applied rewrites 95.5%

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

Alternative 7: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

5. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-log.f64 95.4

      \[\leadsto \frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\log \left(0.25 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\log f}}} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{1}{\frac{0.25 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\log \left(0.25 \cdot \mathsf{PI}\left(\right)\right) + \log f}}} \]

9. Applied rewrites 95.3%

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

Reproduce

herbie shell --seed 2024302 
(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)))))))))