VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 99.0%

Time: 16.0s

Alternatives: 8

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1064647126.2044886

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.125\\ t_1 := \tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathsf{fma}\left(t\_0, \log t\_1, \log \left({t\_1}^{t\_0}\right)\right) \cdot \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (PI) 0.125)) (t_1 (tanh (* (* f 0.25) (PI)))))
   (* (fma t_0 (log t_1) (log (pow t_1 t_0))) (/ 4.0 (* (* (PI) (PI)) 0.25)))))

Derivation

1. Initial program 5.4%

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \color{blue}{\frac{1}{\frac{1}{-\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

6. Applied rewrites 98.9%

   \[\leadsto -\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}}\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub0-neg N/A

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

   6. frac-2neg-rev N/A

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

8. Applied rewrites 99.1%

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

   1. lift-log.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. log-prod N/A

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

   5. log-pow N/A

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

   6. lift-log.f64 N/A

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

   7. lower-fma.f64 N/A

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

10. Applied rewrites 99.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.125, \log \tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right), \log \left({\tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\mathsf{PI}\left(\right) \cdot 0.125\right)}\right)\right)} \cdot \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25} \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \log \left({\tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\mathsf{PI}\left(\right) \cdot 0.125\right)}\right)\right) \cdot \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (* 2.0 (log (pow (tanh (* (* f 0.25) (PI))) (* (PI) 0.125))))
  (/ 4.0 (* (* (PI) (PI)) 0.25))))

Derivation

1. Initial program 5.4%

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \color{blue}{\frac{1}{\frac{1}{-\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

6. Applied rewrites 98.9%

   \[\leadsto -\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}}\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub0-neg N/A

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

   6. frac-2neg-rev N/A

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

8. Applied rewrites 99.1%

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

   1. lift-log.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. log-pow N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

10. Applied rewrites 99.1%

    \[\leadsto \color{blue}{\left(2 \cdot \log \left({\tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\mathsf{PI}\left(\right) \cdot 0.125\right)}\right)\right)} \cdot \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25} \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI))))
   (* (log (pow (tanh (* t_0 f)) t_0)) (/ 4.0 (* (* (PI) (PI)) 0.25)))))

Derivation

1. Initial program 5.4%

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \color{blue}{\frac{1}{\frac{1}{-\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

6. Applied rewrites 98.9%

   \[\leadsto -\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}}\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub0-neg N/A

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

   6. frac-2neg-rev N/A

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

8. Applied rewrites 99.1%

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

Alternative 4: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-16}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (* -0.25 (PI))
  (* (log (tanh (* (* f 0.25) (PI)))) (/ -16.0 (* (PI) (PI))))))

Derivation

1. Initial program 5.4%

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 98.7%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \color{blue}{\frac{1}{\frac{1}{-\log \tanh \left(f \cdot \left(0.25 \cdot \mathsf{PI}\left(\right)\right)\right)}}} \]

6. Applied rewrites 98.9%

   \[\leadsto -\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)}}\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - \left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot 4\right)}}{\mathsf{neg}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub0-neg N/A

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

   6. frac-2neg-rev N/A

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

8. Applied rewrites 99.1%

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

10. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(f \cdot 0.25\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-16}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right)} \]
11. Add Preprocessing

Alternative 5: 98.9% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.4%

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

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

   6. clear-num-rev N/A

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

4. Applied rewrites 99.0%

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

Alternative 6: 96.3% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.4%

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

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

   6. clear-num-rev N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-log.f64 95.5

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

7. Applied rewrites 95.5%

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

Alternative 7: 96.3% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.4%

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

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

   6. clear-num-rev N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 95.4

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

7. Applied rewrites 95.4%

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

Alternative 8: 96.2% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.4%

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

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

   6. clear-num-rev N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 95.4

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

7. Applied rewrites 95.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. clear-num-rev N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 95.2

      \[\leadsto \color{blue}{\log \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right) \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{4}}} \]

9. Applied rewrites 95.2%

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

Reproduce

herbie shell --seed 2024305 
(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)))))))))