VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 97.2%

Time: 10.9s

Alternatives: 4

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1057649921.8464454

Specification

Math

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

FPCore

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.2% 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 9.1%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 97.8%

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

7. Applied rewrites 97.8%

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

Alternative 2: 96.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 9.1%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 97.8%

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

7. Applied rewrites 97.8%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cosh.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sinh.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. log-div N/A

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

   12. lower--.f64 N/A

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

9. Applied rewrites 97.8%

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

10. Taylor expanded in f around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. pow-prod-down N/A

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

    4. lower-pow.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 95.6

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

12. Applied rewrites 95.6%

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

Alternative 3: 95.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot 0.03125, f, 1\right) + 1}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log
   (/
    (+ (fma (* (* (* (PI) (PI)) f) 0.03125) f 1.0) 1.0)
    (* (* (PI) 0.5) f)))))

Derivation

1. Initial program 9.1%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 95.0

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]

5. Applied rewrites 95.0%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}}\right) \]

6. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\left(1 + f \cdot \left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot f}\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right), \color{blue}{f}, 1\right) + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot f}\right) \]

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 95.0

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.03125, 0.25 \cdot \mathsf{PI}\left(\right)\right), f, 1\right) + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]

8. Applied rewrites 95.0%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.03125, 0.25 \cdot \mathsf{PI}\left(\right)\right), f, 1\right)} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]

9. Taylor expanded in f around inf

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\frac{1}{32} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right), f, 1\right) + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot f}\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. pow2 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-*.f64 93.8

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot 0.03125, f, 1\right) + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]

11. Applied rewrites 93.8%

    \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot 0.03125, f, 1\right) + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]

12. Taylor expanded in f around 0

    \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot \frac{1}{32}, f, 1\right) + \color{blue}{1}}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot f}\right) \]
13. Step-by-step derivation

14. Applied rewrites 95.1%

    \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot 0.03125, f, 1\right) + \color{blue}{1}}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]

15. Final simplification 95.1%

    \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f\right) \cdot 0.03125, f, 1\right) + 1}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \]
16. Add Preprocessing

Alternative 4: 95.9% 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 9.1%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 94.9%

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

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 94.9

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

8. Applied rewrites 94.9%

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

Reproduce

herbie shell --seed 2025037 
(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)))))))))