VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 99.4%

Time: 16.5s

Alternatives: 7

Speedup: 4.8×

• could not determine a ground truth

  1. f = 948048012.5371027

Specification

Math

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

FPCore

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{0.5}\right)}^{\left(-f\right)}\\ t_1 := 0.25 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\frac{\mathsf{PI}\left(\right)}{4} \cdot f \leq 0.002:\\ \;\;\;\;\frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot \left(\left(t\_1 \cdot -4\right) \cdot \log \tanh \left(t\_1 \cdot f\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(-t\_0\right) - \mathsf{log1p}\left(t\_0\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (pow (pow (exp (PI)) 0.5) (- f))) (t_1 (* 0.25 (PI))))
   (if (<= (* (/ (PI) 4.0) f) 0.002)
     (* (/ (/ -4.0 (PI)) (PI)) (* (* t_1 -4.0) (log (tanh (* t_1 f)))))
     (/ (- (log1p (- t_0)) (log1p t_0)) t_1))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f) < 2e-3

   1. Initial program 5.7%

      \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
   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. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)}{\left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      10. div-inv N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      11. clear-num N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      13. lower-/.f64 99.5

          \[\leadsto \color{blue}{\frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

   8. Applied rewrites 99.5%

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

   if 2e-3 < (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)

   1. Initial program 9.2%

      \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
   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. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

   4. Applied rewrites 63.2%

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

   6. Applied rewrites 96.5%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(-{\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{0.5}\right)}^{\left(-f\right)}\right) - \mathsf{log1p}\left({\left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{0.5}\right)}^{\left(-f\right)}\right)}}{0.25 \cdot \mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \mathsf{PI}\left(\right)\\ t_1 := f \cdot t\_0\\ t_2 := -\mathsf{PI}\left(\right)\\ t_3 := \frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}\\ t_4 := {\left(e^{-0.5}\right)}^{\left(f \cdot \mathsf{PI}\left(\right)\right)}\\ \mathbf{if}\;\frac{\mathsf{PI}\left(\right)}{4} \cdot f \leq 0.5:\\ \;\;\;\;t\_3 \cdot \mathsf{fma}\left(-\log \cosh t\_1, t\_2, \log \sinh t\_1 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \left(\left(t\_0 \cdot -4\right) \cdot \left(\mathsf{log1p}\left(-t\_4\right) - \mathsf{log1p}\left(t\_4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI)))
        (t_1 (* f t_0))
        (t_2 (- (PI)))
        (t_3 (/ (/ -4.0 (PI)) (PI)))
        (t_4 (pow (exp -0.5) (* f (PI)))))
   (if (<= (* (/ (PI) 4.0) f) 0.5)
     (* t_3 (fma (- (log (cosh t_1))) t_2 (* (log (sinh t_1)) t_2)))
     (* t_3 (* (* t_0 -4.0) (- (log1p (- t_4)) (log1p t_4)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f) < 0.5

   1. Initial program 6.1%

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)}{\left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      10. div-inv N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      11. clear-num N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      13. lower-/.f64 99.5

          \[\leadsto \color{blue}{\frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

   8. Applied rewrites 99.5%

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

   10. Applied rewrites 99.5%

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

   if 0.5 < (*.f64 (/.f64 (PI.f64) #s(literal 4 binary64)) f)

   1. Initial program 1.8%

      \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
   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. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

   4. Applied rewrites 60.1%

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

   6. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)}{\left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      10. div-inv N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      11. clear-num N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

      13. lower-/.f64 60.1

          \[\leadsto \color{blue}{\frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)\right) \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

   8. Applied rewrites 60.1%

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

      1. lift-log.f64 N/A

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

      2. lift-tanh.f64 N/A

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

      3. tanh-def-c N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

   10. Applied rewrites 96.1%

       \[\leadsto \frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot \left(\left(\left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot -4\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(-{\left(e^{-0.5}\right)}^{\left(f \cdot \mathsf{PI}\left(\right)\right)}\right) - \mathsf{log1p}\left({\left(e^{-0.5}\right)}^{\left(f \cdot \mathsf{PI}\left(\right)\right)}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI))))
   (* (/ (/ -4.0 (PI)) (PI)) (* (* t_0 -4.0) (log (tanh (* t_0 f)))))))

Derivation

1. Initial program 5.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)}{\left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}{0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-1}{4} \cdot \mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-1}{4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   10. div-inv N/A

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right)}{-4}}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   11. clear-num N/A

       \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-4}{\mathsf{PI}\left(\right)}}}{\mathsf{PI}\left(\right)} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right) \]

   13. lower-/.f64 97.7

       \[\leadsto \color{blue}{\frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}} \cdot \left(0 \cdot \mathsf{PI}\left(\right) - \left(-0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot 0.25\right) \cdot f\right) \cdot -4\right)\right) \]

   14. lift--.f64 N/A

       \[\leadsto \frac{\frac{-4}{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(0 \cdot \mathsf{PI}\left(\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \tanh \left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot f\right) \cdot -4\right)\right)} \]

8. Applied rewrites 97.7%

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

Alternative 4: 99.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* 0.25 (PI)))) (/ (log (tanh (* f t_0))) t_0)))

Derivation

1. Initial program 5.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 97.7%

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

Alternative 5: 96.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \mathsf{PI}\left(\right), -0.010416666666666666, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (/ -1.0 (/ (PI) 4.0))
  (log
   (/
    (fma
     (* (fma (* 4.0 (PI)) -0.010416666666666666 (* 0.125 (PI))) f)
     f
     (/ 4.0 (PI)))
    f))))

Derivation

1. Initial program 5.9%

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

3. 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 94.6%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-2, \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot 2\right) \cdot 0.005208333333333333, \left(0.0625 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 94.6%

   \[\leadsto -\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \mathsf{PI}\left(\right), -0.010416666666666666, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]

8. Final simplification 94.6%

   \[\leadsto \frac{-1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(4 \cdot \mathsf{PI}\left(\right), -0.010416666666666666, 0.125 \cdot \mathsf{PI}\left(\right)\right) \cdot f, f, \frac{4}{\mathsf{PI}\left(\right)}\right)}{f}\right) \]
9. Add Preprocessing

Alternative 6: 95.8% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 94.2

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

7. Applied rewrites 94.2%

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

Alternative 7: 95.7% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 5.9%

   \[-\frac{1}{\frac{\mathsf{PI}\left(\right)}{4}} \cdot \log \left(\frac{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} + e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}{e^{\frac{\mathsf{PI}\left(\right)}{4} \cdot f} - e^{-\frac{\mathsf{PI}\left(\right)}{4} \cdot f}}\right) \]
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. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in f around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 94.2

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

7. Applied rewrites 94.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 94.0

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

9. Applied rewrites 94.0%

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

Reproduce

herbie shell --seed 2024322 
(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)))))))))