VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 98.9%

Time: 12.4s

Alternatives: 9

Speedup: 4.6×

• could not determine a ground truth

  1. f = 1524768334.5395565

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: 6.8% 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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{4}\\ t_1 := \frac{\mathsf{PI}\left(\right)}{-4}\\ t_2 := e^{t\_0 \cdot f}\\ t_3 := e^{t\_0 \cdot \left(-f\right)}\\ \mathbf{if}\;\frac{1}{t\_0} \cdot \log \left(\frac{t\_2 + t\_3}{t\_2 - t\_3}\right) \leq \infty:\\ \;\;\;\;\frac{\log \left(\frac{\cosh \left(t\_1 \cdot f\right)}{\sinh \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \log \left({\left({\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}^{1}\right)}^{\left({\left(e^{f}\right)}^{t\_1}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0))
        (t_1 (/ (PI) -4.0))
        (t_2 (exp (* t_0 f)))
        (t_3 (exp (* t_0 (- f)))))
   (if (<= (* (/ 1.0 t_0) (log (/ (+ t_2 t_3) (- t_2 t_3)))) INFINITY)
     (* (/ (log (/ (cosh (* t_1 f)) (sinh (* (* (PI) f) 0.25)))) (PI)) -4.0)
     (*
      (/ -1.0 t_0)
      (log (pow (pow (/ 2.0 (* (* (PI) 0.5) f)) 1.0) (pow (exp f) t_1)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (/.f64 (PI.f64) #s(literal 4 binary64))) (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f))))))) < +inf.0

   1. Initial program 6.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in f around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if +inf.0 < (*.f64 (/.f64 #s(literal 1 binary64) (/.f64 (PI.f64) #s(literal 4 binary64))) (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)))))))

   1. Initial program 0.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in f around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 100.0

         \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\left(\frac{2}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot f}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\color{blue}{\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]

   8. Taylor expanded in f around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}^{\color{blue}{1}}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot \left(-f\right)}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot \left(-f\right)}}\right) \leq \infty:\\ \;\;\;\;\frac{\log \left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(\left(\mathsf{PI}\left(\right) \cdot f\right) \cdot 0.25\right)}\right)}{\mathsf{PI}\left(\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}^{1}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)) (t_1 (/ (PI) -4.0)))
   (*
    (/ -1.0 t_0)
    (log
     (pow
      (pow (/ (cosh (* t_1 f)) (sinh (* f t_0))) (pow (exp f) t_0))
      (pow (exp f) t_1))))))

Derivation

1. Initial program 6.8%

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\left(\frac{\cosh \left(\frac{\mathsf{PI}\left(\right)}{-4} \cdot f\right)}{\sinh \left(f \cdot \frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]
6. Add Preprocessing

Alternative 3: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{4}\\ \frac{-1}{t\_0} \cdot \log \left({\left({\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}^{\left({\left(e^{f}\right)}^{t\_0}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (/ (PI) 4.0)))
   (*
    (/ -1.0 t_0)
    (log
     (pow
      (pow (/ 2.0 (* (* (PI) 0.5) f)) (pow (exp f) t_0))
      (pow (exp f) (/ (PI) -4.0)))))))

Derivation

1. Initial program 6.8%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in f around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

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

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 98.0

      \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\left(\frac{2}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 0.5\right) \cdot f}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]

7. Applied rewrites 98.0%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\color{blue}{\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]

8. Final simplification 98.0%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left({\left({\left(\frac{2}{\left(\mathsf{PI}\left(\right) \cdot 0.5\right) \cdot f}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{4}\right)}\right)}\right)}^{\left({\left(e^{f}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{-4}\right)}\right)}\right) \]
9. Add Preprocessing

Alternative 4: 97.1% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ (log (/ (cosh (* (/ (PI) -4.0) f)) (sinh (* (* (PI) f) 0.25)))) (PI))
  -4.0))

Derivation

1. Initial program 6.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 97.2

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

7. Applied rewrites 97.2%

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

Alternative 5: 96.4% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.8%

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

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{1}{\mathsf{PI}\left(\right)}, 4, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{\mathsf{PI}\left(\right)}^{3}}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot 0.020833333333333332, -2, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot 0.125\right), f, 0\right) \cdot f\right)}{f}\right)} \]

7. Applied rewrites 96.8%

   \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{\mathsf{PI}\left(\right)}^{3}}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}, -0.041666666666666664, 0.125 \cdot \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)}\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)}{\frac{\mathsf{PI}\left(\right)}{4}}} \]

8. Taylor expanded in f around 0

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

10. Applied rewrites 96.8%

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

11. Final simplification 96.8%

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

Alternative 6: 96.3% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.8%

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

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{1}{\mathsf{PI}\left(\right)}, 4, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{\mathsf{PI}\left(\right)}^{3}}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot 0.020833333333333332, -2, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot 0.125\right), f, 0\right) \cdot f\right)}{f}\right)} \]

6. Taylor expanded in f around inf

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

8. Applied rewrites 50.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f64 50.5

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

10. Applied rewrites 50.5%

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

11. Taylor expanded in f around 0

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

13. Applied rewrites 96.7%

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

14. Final simplification 96.7%

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

Alternative 7: 95.9% accurate, 4.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.8%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. metadata-eval N/A

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

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

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

   8. mul-1-neg N/A

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

   9. lower-/.f64 N/A

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

   10. mul-1-neg N/A

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

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

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

   12. metadata-eval N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

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

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

   17. lower-*.f64 N/A

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

7. Applied rewrites 96.3%

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

Alternative 8: 95.7% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 6.8%

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 96.1

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

5. Applied rewrites 96.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 96.1%

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

9. Applied rewrites 96.1%

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

10. Final simplification 96.1%

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

Alternative 9: 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 6.8%

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

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{1}{\mathsf{PI}\left(\right)}, 4, \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{{\mathsf{PI}\left(\right)}^{3}}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot 0.020833333333333332, -2, \frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right)} \cdot 0.125\right), f, 0\right) \cdot f\right)}{f}\right)} \]

6. Taylor expanded in f around inf

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

8. Applied rewrites 50.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f64 50.5

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

10. Applied rewrites 50.5%

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

11. Taylor expanded in f around inf

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

13. Applied rewrites 1.6%

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

14. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2025017 
(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)))))))))