VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 97.0%

Time: 11.5s

Alternatives: 6

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1317562078.9234354

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: 6.6% 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 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: 97.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (* f (PI)) 0.25)))
   (* (/ (log (/ (cosh t_0) (sinh t_0))) (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 98.3%

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

7. Applied rewrites 98.3%

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

Alternative 2: 96.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \mathsf{PI}\left(\right)\\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, t\_0\right)\right), f, 2\right)}{\left(\mathsf{fma}\left(f \cdot f, {\mathsf{PI}\left(\right)}^{3} \cdot 0.005208333333333333, t\_0\right) - -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI))))
   (*
    (/ -1.0 (/ (PI) 4.0))
    (log
     (/
      (fma (fma -0.25 (PI) (fma (* (* (PI) (PI)) f) 0.0625 t_0)) f 2.0)
      (*
       (-
        (fma (* f f) (* (pow (PI) 3.0) 0.005208333333333333) t_0)
        (* -0.25 (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

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

5. Applied rewrites 4.5%

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

6. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{2 + f \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{16} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - 1}\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 4.1%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - 1}\right) \]

9. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\color{blue}{f \cdot \left(\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) + {f}^{2} \cdot \left(\frac{1}{384} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{384} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}}\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

11. Applied rewrites 97.2%

    \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(f \cdot f, {\mathsf{PI}\left(\right)}^{3} \cdot 0.005208333333333333, 0.25 \cdot \mathsf{PI}\left(\right)\right) - -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f}}\right) \]

12. Final simplification 97.2%

    \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\left(\mathsf{fma}\left(f \cdot f, {\mathsf{PI}\left(\right)}^{3} \cdot 0.005208333333333333, 0.25 \cdot \mathsf{PI}\left(\right)\right) - -0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot f}\right) \]
13. Add Preprocessing

Alternative 3: 96.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log
   (/
    (fma (fma -0.25 (PI) (fma (* (* (PI) (PI)) f) 0.0625 (* 0.25 (PI)))) f 2.0)
    (* (sinh (* (* f (PI)) 0.25)) 2.0)))))

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

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

5. Applied rewrites 4.5%

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

6. Taylor expanded in f around 0

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{2 + f \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \left(\frac{1}{16} \cdot \left(f \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - 1}\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 4.1%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - 1}\right) \]

9. Taylor expanded in f around inf

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\color{blue}{e^{\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)} - e^{\mathsf{neg}\left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}}\right) \]
10. Step-by-step derivation

    1. sinh-undef N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{2 \cdot \color{blue}{\sinh \left(\frac{1}{4} \cdot \left(f \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \]

    2. *-commutative N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{2 \cdot \sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot \color{blue}{2}}\right) \]

    4. lift-*.f64 N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot 2}\right) \]

    5. lift-PI.f64 N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot 2}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot 2}\right) \]

    7. lift-sinh.f64 N/A

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, \frac{1}{16}, \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}\right) \cdot 2}\right) \]

    8. lift-*.f64 97.1

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot \color{blue}{2}}\right) \]

11. Applied rewrites 97.1%

    \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\color{blue}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2}}\right) \]

12. Final simplification 97.1%

    \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right), \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.0625, 0.25 \cdot \mathsf{PI}\left(\right)\right)\right), f, 2\right)}{\sinh \left(\left(f \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 2}\right) \]
13. Add Preprocessing

Alternative 4: 96.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := f \cdot \mathsf{PI}\left(\right)\\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{t\_0 \cdot 0.25} + \mathsf{fma}\left(t\_0, -0.25, 1\right)}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* f (PI))))
   (*
    (/ -1.0 (/ (PI) 4.0))
    (log (/ (+ (exp (* t_0 0.25)) (fma t_0 -0.25 1.0)) (* (* (PI) 0.5) 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

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

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 96.8

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

8. Applied rewrites 96.8%

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

9. Taylor expanded in f around 0

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

    1. *-commutative N/A

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

    2. lift-*.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-*.f64 96.8

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

11. Applied rewrites 96.8%

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

12. Final simplification 96.8%

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

Alternative 5: 96.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot f, 0.03125, 0.25 \cdot \mathsf{PI}\left(\right)\right), f, 1\right) + \mathsf{fma}\left(f \cdot \mathsf{PI}\left(\right), -0.25, 1\right)}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log
   (/
    (+
     (fma (fma (* (* (PI) (PI)) f) 0.03125 (* 0.25 (PI))) f 1.0)
     (fma (* f (PI)) -0.25 1.0))
    (* (* (PI) 0.5) 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

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

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in f around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 96.8

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

8. Applied rewrites 96.8%

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

9. Taylor expanded in f around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. lift-PI.f64 N/A

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

    12. lower-*.f64 N/A

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

    13. lift-PI.f64 96.8

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

11. Applied rewrites 96.8%

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

12. Final simplification 96.8%

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

Alternative 6: 96.0% 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 96.7%

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

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 96.7

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

8. Applied rewrites 96.7%

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

9. Final simplification 96.7%

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

Reproduce

herbie shell --seed 2025051 
(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)))))))))