VandenBroeck and Keller, Equation (20)

Percentage Accurate: 18.6% → 96.9%

Time: 9.3s

Alternatives: 5

Speedup: 4.8×

• could not determine a ground truth

  1. f = 965186816.0

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 binary32
 (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: 18.6% 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 binary32
 (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: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \mathsf{PI}\left(\right)\\ t_1 := \sinh \left(t\_0 \cdot 0.25\right)\\ t_2 := \log t\_1\\ t_3 := 0.25 \cdot t\_0\\ t_4 := \log \cosh \left(t\_0 \cdot -0.25\right)\\ \mathbf{if}\;f \leq 10:\\ \;\;\;\;\frac{\log \left(\frac{\cosh t\_3}{\sinh t\_3}\right) \cdot -4}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-{\log \left(f \cdot \left(0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)\right)\right)}^{3}}{\mathsf{fma}\left(t\_4, t\_4, \mathsf{fma}\left(t\_2, t\_2, \log \left({t\_1}^{t\_4}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary32
 (let* ((t_0 (* f (PI)))
        (t_1 (sinh (* t_0 0.25)))
        (t_2 (log t_1))
        (t_3 (* 0.25 t_0))
        (t_4 (log (cosh (* t_0 -0.25)))))
   (if (<= f 10.0)
     (/ (* (log (/ (cosh t_3) (sinh t_3))) -4.0) (PI))
     (/
      (*
       (/
        (- (pow (log (* f (* 0.5 (* (PI) 0.5)))) 3.0))
        (fma t_4 t_4 (fma t_2 t_2 (log (pow t_1 t_4)))))
       -4.0)
      (PI)))))

Derivation

1. Split input into 2 regimes

2. if f < 10

   1. Initial program 22.9%

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

   3. Taylor expanded in f around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

   if 10 < f

   1. Initial program 5.2%

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

   3. Taylor expanded in f around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 5.2%

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

      1. lift-log.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-cosh.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sinh.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. diff-log N/A

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

   9. Applied rewrites 5.2%

      \[\leadsto \frac{\frac{{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right)}^{3} - {\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}^{3}}{\mathsf{fma}\left(\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right), \log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right), \mathsf{fma}\left(\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \log \left({\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}^{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right)}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} \]

   10. Taylor expanded in f around 0

       \[\leadsto \frac{\frac{-1 \cdot {\left(\log f + \log \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\mathsf{fma}\left(\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \mathsf{fma}\left(\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \left({\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}^{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right)}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\frac{\mathsf{neg}\left({\left(\log f + \log \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}\right)}{\mathsf{fma}\left(\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \mathsf{fma}\left(\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \left({\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}^{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right)}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} \]

       2. lower-neg.f32 N/A

          \[\leadsto \frac{\frac{-{\left(\log f + \log \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\mathsf{fma}\left(\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \mathsf{fma}\left(\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \left({\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}^{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right)}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} \]

       3. lower-pow.f32 N/A

          \[\leadsto \frac{\frac{-{\left(\log f + \log \left(\frac{1}{2} \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\mathsf{fma}\left(\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right), \mathsf{fma}\left(\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right), \log \left({\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}^{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{4}\right)}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 95.6%

       \[\leadsto \frac{\frac{-{\log \left(f \cdot \left(0.5 \cdot \left(\mathsf{PI}\left(\right) \cdot 0.5\right)\right)\right)}^{3}}{\mathsf{fma}\left(\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right), \log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right), \mathsf{fma}\left(\log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right), \log \left({\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}^{\log \cosh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right)}\right)\right)\right)} \cdot -4}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary32
 (let* ((t_0 (* 0.25 (* f (PI)))))
   (/ (* (log (/ (cosh t_0) (sinh t_0))) -4.0) (PI))))

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 82.7%

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

7. Applied rewrites 82.7%

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

Alternative 3: 81.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary32
 (let* ((t_0 (* f (PI))))
   (* (log (/ (cosh (* t_0 -0.25)) (sinh (* t_0 0.25)))) (/ -4.0 (PI)))))

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 82.7%

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

7. Applied rewrites 82.7%

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

9. Applied rewrites 82.4%

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

Alternative 4: 80.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333, f \cdot f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary32
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log (/ (fma (* (PI) 0.08333333333333333) (* f f) (/ 4.0 (PI))) f))))

Derivation

1. Initial program 20.0%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 80.2%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{2}{\mathsf{PI}\left(\right)}, 2, \mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot 0.5}, 0, \mathsf{fma}\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)}, 0.125, -2 \cdot \frac{{\mathsf{PI}\left(\right)}^{3} \cdot 0.005208333333333333}{{\left(\mathsf{PI}\left(\right) \cdot 0.5\right)}^{2}}\right) \cdot f\right) \cdot f\right)}{f}\right)} \]

6. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{{f}^{2} \cdot \left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) + 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{f}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right)\right) \cdot {f}^{2} + 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}}{f}\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{-1}{24} \cdot \mathsf{PI}\left(\right) + \frac{1}{8} \cdot \mathsf{PI}\left(\right), {f}^{2}, 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

   3. distribute-rgt-out N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right), {f}^{2}, 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right), {f}^{2}, 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

   5. lift-PI.f32 N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{24} + \frac{1}{8}\right), {f}^{2}, 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

   6. metadata-eval N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \frac{1}{12}, {f}^{2}, 4 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lift-PI.f32 80.2

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333, f \cdot f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

8. Applied rewrites 80.2%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333, f \cdot f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

9. Final simplification 80.2%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot 0.08333333333333333, f \cdot f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]
10. Add Preprocessing

Alternative 5: 79.2% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary32
 (* (/ (- (log (* 0.25 (* f (PI))))) (PI)) -4.0))

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 82.7%

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

7. Applied rewrites 82.6%

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

8. Taylor expanded in f around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. +-commutative N/A

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

   4. sum-log N/A

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

   5. lower-log.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 78.3

       \[\leadsto \frac{-\log \left(\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

10. Applied rewrites 78.3%

    \[\leadsto \frac{-\log \left(\left(\left(0.5 \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5\right) \cdot f\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

11. Taylor expanded in f around 0

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

    1. lower-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-PI.f32 78.3

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

13. Applied rewrites 78.3%

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

Reproduce

herbie shell --seed 2025054 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary32
  (- (* (/ 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)))))))))