VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.8%

Time: 16.3s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1005637934.0836483

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: 98.8% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 99.1%

   \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
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-PI.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 99.1

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 99.1%

   \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
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-PI.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 99.1

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 99.1

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

8. Applied rewrites 99.1%

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

9. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 99.1%

   \[\leadsto \color{blue}{{\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \left({\left(\frac{4}{\mathsf{PI}\left(\right)}\right)}^{0.5} \cdot \log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
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 99.1%

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

7. Final simplification 99.1%

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

Alternative 4: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 99.1%

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

5. Final simplification 99.1%

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

Alternative 5: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.7%

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

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

4. Applied rewrites 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 95.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)}{-0.25 \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log (/ (fma (* (* 0.08333333333333333 (PI)) f) f (/ 4.0 (PI))) f))
  (* -0.25 (PI))))

Derivation

1. Initial program 6.7%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 95.4%

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

6. Taylor expanded in f around 0

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

8. Applied rewrites 95.4%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]
9. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{12} \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\frac{1}{12} \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)}\right) \]

10. Applied rewrites 95.5%

    \[\leadsto \color{blue}{\frac{\log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)}{-0.25 \cdot \mathsf{PI}\left(\right)}} \]
11. Add Preprocessing

Alternative 7: 95.4% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.4%

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

9. Applied rewrites 95.4%

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

10. Final simplification 95.4%

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

Reproduce

herbie shell --seed 2024255 
(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)))))))))