VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 97.1%

Time: 11.4s

Alternatives: 6

Speedup: 4.6×

• could not determine a ground truth

  1. f = 970991337.8746042

Specification

Math

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

FPCore

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f (PI)) 0.25)))
   (/ (* (log (/ (cosh t_0) (sinh t_0))) -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. 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-*.f64 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 96.7%

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

7. Applied rewrites 96.7%

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

Alternative 2: 96.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \mathsf{PI}\left(\right)\\ \frac{{t\_0}^{2} \cdot 0.03125 - \log \sinh \left(t\_0 \cdot 0.25\right)}{\mathsf{PI}\left(\right)} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (PI))))
   (* (/ (- (* (pow t_0 2.0) 0.03125) (log (sinh (* t_0 0.25)))) (PI)) -4.0)))

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. 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-*.f64 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 96.7%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in f around 0

   \[\leadsto \frac{\frac{1}{32} \cdot \left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left({f}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

   3. pow-prod-down N/A

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot \frac{1}{32} - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

   6. lift-PI.f64 96.2

      \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 0.03125 - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

10. Applied rewrites 96.2%

    \[\leadsto \frac{{\left(f \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot 0.03125 - \log \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]
11. Add Preprocessing

Alternative 3: 96.2% 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 binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log (/ (fma (* (PI) 0.08333333333333333) (* f f) (/ 4.0 (PI))) 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 96.1%

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

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.f64 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-*.f64 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.f64 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-*.f64 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-/.f64 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.f64 96.1

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

8. Applied rewrites 96.1%

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

9. Final simplification 96.1%

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

Alternative 4: 95.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right) \cdot -4}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* (log (/ (/ 2.0 (* 0.5 (PI))) f)) -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. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\log \left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}{\mathsf{PI}\left(\right)} \cdot -4} \]
6. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. log-div N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-log.f64 N/A

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

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

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 95.6

       \[\leadsto \frac{\log \left(\frac{2}{f}\right) - \log \left(0.5 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

7. Applied rewrites 95.6%

   \[\leadsto \frac{\log \left(\frac{2}{f}\right) - \log \left(0.5 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)} \cdot -4 \]

9. Applied rewrites 95.7%

   \[\leadsto \frac{\log \left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right) \cdot -4}{\color{blue}{\mathsf{PI}\left(\right)}} \]
10. Add Preprocessing

Alternative 5: 95.8% accurate, 4.6× speedup

Math

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

FPCore

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

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. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 95.7

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

8. Applied rewrites 95.7%

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

Alternative 6: 1.6% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (* (/ (- 4.0) (PI)) (log (* (* 0.08333333333333333 (PI)) 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 96.1%

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

6. Taylor expanded in f around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. metadata-eval 1.7

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

8. Applied rewrites 1.7%

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

   1. lift-neg.f64 N/A

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

10. Applied rewrites 1.7%

    \[\leadsto \color{blue}{\frac{4}{\mathsf{PI}\left(\right)} \cdot \left(-\log \left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f\right)\right)} \]

11. Final simplification 1.7%

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

Reproduce

herbie shell --seed 2025073 
(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)))))))))