VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 97.1%

Time: 14.9s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 1102439019.5929773

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: 97.1% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 98.8

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

9. Applied rewrites 98.8%

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

11. Applied rewrites 98.8%

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

Alternative 2: 97.1% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 98.8

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

9. Applied rewrites 98.8%

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

Alternative 3: 97.1% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 4: 96.4% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{f \cdot \left(\frac{-1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} + f \cdot \left(\frac{1}{16} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)} - 2 \cdot \frac{\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}}{{\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right)\right)\right) + 2 \cdot \frac{1}{\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]

5. Applied rewrites 98.0%

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

6. Final simplification 98.0%

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

Alternative 5: 96.0% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out-- N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 97.8

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

9. Applied rewrites 97.8%

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

10. Final simplification 97.8%

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

Alternative 6: 95.9% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

4. Applied rewrites 83.0%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out-- N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 82.0

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

7. Applied rewrites 82.0%

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

   1. lift-log.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. log-pow N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 97.6

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

9. Applied rewrites 97.6%

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

Alternative 7: 1.6% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.3%

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

4. Applied rewrites 83.0%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out-- N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 82.0

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

7. Applied rewrites 82.0%

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

   1. lift-log.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. log-pow N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 97.6

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

9. Applied rewrites 97.6%

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

11. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024361 
(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)))))))))