VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.0%

Time: 17.2s

Alternatives: 4

Speedup: 4.8×

• could not determine a ground truth

  1. f = 961857993.6086422

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: 99.0% 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 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. frac-2neg N/A

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

4. Applied rewrites 99.3%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. frac-2neg-rev N/A

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

   11. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 96.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{PI}\left(\right)\\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \frac{0.005208333333333333}{t\_0} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 0.0625 \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f, f, \frac{2}{t\_0}\right)}{f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.5 (PI))))
   (*
    (/ -1.0 (/ (PI) 4.0))
    (log
     (/
      (fma
       (*
        (fma
         -2.0
         (* (/ 0.005208333333333333 t_0) (* 2.0 (* (PI) (PI))))
         (* 0.0625 (* 2.0 (PI))))
        f)
       f
       (/ 2.0 t_0))
      f)))))

Derivation

1. Initial program 7.0%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 96.4%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \frac{0.005208333333333333}{0.5 \cdot \mathsf{PI}\left(\right)}, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 0.0625\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right)} \]

6. Final simplification 96.4%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \frac{0.005208333333333333}{0.5 \cdot \mathsf{PI}\left(\right)} \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 0.0625 \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right) \]
7. Add Preprocessing

Alternative 3: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. frac-2neg N/A

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

4. Applied rewrites 99.3%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift-/.f64 N/A

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

   10. frac-2neg-rev N/A

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

   11. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 96.2

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

9. Applied rewrites 96.2%

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

10. Final simplification 96.2%

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

Alternative 4: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. frac-2neg N/A

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 96.0

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

7. Applied rewrites 96.0%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 96.0

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

9. Applied rewrites 96.0%

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

10. Final simplification 96.0%

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

Reproduce

herbie shell --seed 2024298 
(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)))))))))