VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.5% → 99.1%

Time: 13.9s

Alternatives: 4

Speedup: 4.6×

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: 6.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. frac-2neg N/A

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 96.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log (/ (fma (* (* 0.08333333333333333 (PI)) f) f (/ 2.0 (* 0.5 (PI)))) f))))

Derivation

1. Initial program 7.8%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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 96.7%

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

6. Taylor expanded in f around 0

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

8. Applied rewrites 96.7%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right) \]

9. Final simplification 96.7%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\left(0.08333333333333333 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}\right)}{f}\right) \]
10. Add Preprocessing

Alternative 3: 96.0% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.8%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 95.9%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num-rev N/A

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

   4. lower-/.f64 95.9

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

7. Applied rewrites 95.9%

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

9. Applied rewrites 95.9%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

11. Applied rewrites 96.1%

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

12. Final simplification 96.1%

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

Alternative 4: 95.8% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 7.8%

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

3. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 95.9%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\frac{2}{0.5 \cdot \mathsf{PI}\left(\right)}}{f}\right)} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num-rev N/A

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

   4. lower-/.f64 95.9

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

7. Applied rewrites 95.9%

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

9. Applied rewrites 95.9%

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

10. Final simplification 95.9%

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

Reproduce

herbie shell --seed 2024312 
(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)))))))))