NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 8.2s

Alternatives: 7

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ (PI) 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

Initial Program: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ (PI) 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

Alternative 1: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-\mathsf{PI}\left(\right)}{a}}{b}}{-2 \cdot \left(b + a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (/ (/ (- (PI)) a) b) (* -2.0 (+ b a))))

Derivation

1. Initial program 81.4%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   11. *-lft-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   13. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   17. +-commutative N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   18. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   20. *-lft-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   21. *-rgt-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

4. Applied rewrites 88.4%

   \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{2 \cdot \left(a + b\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{b - a}}}{2 \cdot \left(a + b\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(a + b\right)}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{-\left(b - a\right)}{b \cdot a} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-2 \cdot \left(b + a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}}{-2 \cdot \left(b + a\right)} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b}}\right)}{-2 \cdot \left(b + a\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}{b}}}{-2 \cdot \left(b + a\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a}}}{b}}{-2 \cdot \left(b + a\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a}}{b}}}{-2 \cdot \left(b + a\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{a}}}{b}}{-2 \cdot \left(b + a\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{a}}}{b}}{-2 \cdot \left(b + a\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{a}}{b}}{-2 \cdot \left(b + a\right)} \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{-\mathsf{PI}\left(\right)}}{a}}{b}}{-2 \cdot \left(b + a\right)} \]

   10. lower-PI.f64 99.7

       \[\leadsto \frac{\frac{\frac{-\color{blue}{\mathsf{PI}\left(\right)}}{a}}{b}}{-2 \cdot \left(b + a\right)} \]

9. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\frac{\frac{-\mathsf{PI}\left(\right)}{a}}{b}}}{-2 \cdot \left(b + a\right)} \]
10. Add Preprocessing

Alternative 2: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-42} \lor \neg \left(a \leq 4.2 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.6e-42) (not (<= a 4.2e-48)))
   (/ (* 0.5 (PI)) (* (* a b) a))
   (* (/ (PI) (* (* a b) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -9.60000000000000011e-42 or 4.19999999999999977e-48 < a

   1. Initial program 81.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}}{b} \cdot \frac{1}{2} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{a}^{2}}}{b} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot \frac{1}{2} \]

      8. lower-*.f64 77.3

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot 0.5 \]

   5. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.3%

      \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b}}{\color{blue}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.4%

      \[\leadsto \frac{0.5 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(a \cdot b\right) \cdot a}} \]

   if -9.60000000000000011e-42 < a < 4.19999999999999977e-48

   1. Initial program 81.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      21. *-rgt-identity N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \cdot \frac{1}{2} \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)}}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot \frac{1}{2} \]

      8. lower-*.f64 76.5

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot 0.5 \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.1%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-42} \lor \neg \left(a \leq 4.2 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{t\_0}{a \cdot b}}{a}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(a \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (PI))))
   (if (<= a -9.6e-42)
     (/ (/ t_0 (* a b)) a)
     (if (<= a 4.2e-48)
       (* (/ (PI) (* (* a b) b)) 0.5)
       (/ t_0 (* (* a b) a))))))

Derivation

1. Split input into 3 regimes

2. if a < -9.60000000000000011e-42

   1. Initial program 81.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}}{b} \cdot \frac{1}{2} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{a}^{2}}}{b} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot \frac{1}{2} \]

      8. lower-*.f64 79.2

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot 0.5 \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.6%

      \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b}}{\color{blue}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.7%

      \[\leadsto \frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a} \]

   if -9.60000000000000011e-42 < a < 4.19999999999999977e-48

   1. Initial program 81.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      21. *-rgt-identity N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \cdot \frac{1}{2} \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)}}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot \frac{1}{2} \]

      8. lower-*.f64 76.5

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot 0.5 \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.1%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5 \]

   if 4.19999999999999977e-48 < a

   1. Initial program 81.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}}{b} \cdot \frac{1}{2} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{a}^{2}}}{b} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot \frac{1}{2} \]

      8. lower-*.f64 75.8

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot 0.5 \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.7%

      \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b}}{\color{blue}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 85.8%

      \[\leadsto \frac{0.5 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(a \cdot b\right) \cdot a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.6e-42)
   (/ (* (/ (PI) a) 0.5) (* b a))
   (if (<= a 4.2e-48)
     (* (/ (PI) (* (* a b) b)) 0.5)
     (/ (* 0.5 (PI)) (* (* a b) a)))))

Derivation

1. Split input into 3 regimes

2. if a < -9.60000000000000011e-42

   1. Initial program 81.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}}{b} \cdot \frac{1}{2} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{a}^{2}}}{b} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot \frac{1}{2} \]

      8. lower-*.f64 79.2

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot 0.5 \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.6%

      \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{\color{blue}{b \cdot a}} \]

   if -9.60000000000000011e-42 < a < 4.19999999999999977e-48

   1. Initial program 81.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      9. difference-of-squares N/A

         \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      11. *-lft-identity N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      17. +-commutative N/A

          \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      18. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      20. *-lft-identity N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

      21. *-rgt-identity N/A

          \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \cdot \frac{1}{2} \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)}}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot \frac{1}{2} \]

      8. lower-*.f64 76.5

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot 0.5 \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.1%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5 \]

   if 4.19999999999999977e-48 < a

   1. Initial program 81.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b} \cdot \frac{1}{2}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}{b}} \cdot \frac{1}{2} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{{a}^{2}}}}{b} \cdot \frac{1}{2} \]

      6. lower-PI.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{{a}^{2}}}{b} \cdot \frac{1}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot \frac{1}{2} \]

      8. lower-*.f64 75.8

         \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\color{blue}{a \cdot a}}}{b} \cdot 0.5 \]

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot a}}{b} \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.7%

      \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b}}{\color{blue}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 85.8%

      \[\leadsto \frac{0.5 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(a \cdot b\right) \cdot a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-\mathsf{PI}\left(\right)}{a \cdot b}}{-2 \cdot \left(b + a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (/ (- (PI)) (* a b)) (* -2.0 (+ b a))))

Derivation

1. Initial program 81.4%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   11. *-lft-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   13. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   17. +-commutative N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   18. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   20. *-lft-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   21. *-rgt-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

4. Applied rewrites 88.4%

   \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{2 \cdot \left(a + b\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{b - a}}}{2 \cdot \left(a + b\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(a + b\right)}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{-\left(b - a\right)}{b \cdot a} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-2 \cdot \left(b + a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)}}{-2 \cdot \left(b + a\right)} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b}}\right)}{-2 \cdot \left(b + a\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{a}\right)}{b}}}{-2 \cdot \left(b + a\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a}}}{b}}{-2 \cdot \left(b + a\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a}}{b}}}{-2 \cdot \left(b + a\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{a}}}{b}}{-2 \cdot \left(b + a\right)} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{a}}}{b}}{-2 \cdot \left(b + a\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{a}}{b}}{-2 \cdot \left(b + a\right)} \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{-\mathsf{PI}\left(\right)}}{a}}{b}}{-2 \cdot \left(b + a\right)} \]

   10. lower-PI.f64 99.7

       \[\leadsto \frac{\frac{\frac{-\color{blue}{\mathsf{PI}\left(\right)}}{a}}{b}}{-2 \cdot \left(b + a\right)} \]

9. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\frac{\frac{-\mathsf{PI}\left(\right)}{a}}{b}}}{-2 \cdot \left(b + a\right)} \]

10. Taylor expanded in a around 0

    \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]

    3. mul-1-neg N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}}{a \cdot b}}{-2 \cdot \left(b + a\right)} \]

    4. lower-neg.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{-\mathsf{PI}\left(\right)}}{a \cdot b}}{-2 \cdot \left(b + a\right)} \]

    5. lower-PI.f64 N/A

       \[\leadsto \frac{\frac{-\color{blue}{\mathsf{PI}\left(\right)}}{a \cdot b}}{-2 \cdot \left(b + a\right)} \]

    6. lower-*.f64 99.7

       \[\leadsto \frac{\frac{-\mathsf{PI}\left(\right)}{\color{blue}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]

12. Applied rewrites 99.7%

    \[\leadsto \frac{\color{blue}{\frac{-\mathsf{PI}\left(\right)}{a \cdot b}}}{-2 \cdot \left(b + a\right)} \]
13. Add Preprocessing

Alternative 6: 62.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ (PI) (* (* a b) b)) 0.5))

Derivation

1. Initial program 81.4%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   11. *-lft-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   13. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   17. +-commutative N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   18. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   20. *-lft-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   21. *-rgt-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

4. Applied rewrites 88.4%

   \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \cdot \frac{1}{2} \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)}}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot \frac{1}{2} \]

   8. lower-*.f64 56.3

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot 0.5 \]

7. Applied rewrites 56.3%

   \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 62.9%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5 \]
10. Add Preprocessing

Alternative 7: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (PI) (/ 0.5 (* (* a b) b))))

Derivation

1. Initial program 81.4%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \color{blue}{\frac{1}{b \cdot b - a \cdot a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 1}{2 \cdot \left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}{2 \cdot \left(b \cdot b - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   6. lift--.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   9. difference-of-squares N/A

      \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(b - a\right)\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   10. associate-*r* N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\color{blue}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   11. *-lft-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(\color{blue}{1 \cdot b} - a\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \frac{1 \cdot \mathsf{PI}\left(\right)}{\left(2 \cdot \left(b + a\right)\right) \cdot \left(1 \cdot b - \color{blue}{a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   13. times-frac N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(b + a\right)} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{\color{blue}{2 \cdot \left(b + a\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   17. +-commutative N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   18. lower-+.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \color{blue}{\left(a + b\right)}} \cdot \frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{1 \cdot b - a \cdot 1}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   20. *-lft-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{b} - a \cdot 1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

   21. *-rgt-identity N/A

       \[\leadsto \left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - \color{blue}{a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

4. Applied rewrites 88.4%

   \[\leadsto \color{blue}{\left(\frac{1}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}} \cdot \frac{1}{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \cdot \frac{1}{2} \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)}}{a \cdot {b}^{2}} \cdot \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{{b}^{2} \cdot a}} \cdot \frac{1}{2} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot \frac{1}{2} \]

   8. lower-*.f64 56.3

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(b \cdot b\right)} \cdot a} \cdot 0.5 \]

7. Applied rewrites 56.3%

   \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 56.3%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{0.5}{\left(b \cdot b\right) \cdot a}} \]
10. Step-by-step derivation

11. Applied rewrites 62.9%

    \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot \color{blue}{b}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ (PI) 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))