NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.9% → 99.7%

Time: 6.2s

Alternatives: 6

Speedup: 2.4×

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: 77.9% 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, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift--.f64 N/A

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

   10. difference-of-squares N/A

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

   11. lower-/.f64 N/A

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

   12. difference-of-squares N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 86.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 86.5

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

6. Applied rewrites 86.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-inverses N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. *-rgt-identity N/A

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

   9. lower-/.f64 99.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23} \lor \neg \left(a \leq 1.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.6e-23) (not (<= a 1.9e-33)))
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b a) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.59999999999999988e-23 or 1.89999999999999997e-33 < a

   1. Initial program 73.9%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.2

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

   7. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 90.3

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

   9. Applied rewrites 90.3%

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

   if -1.59999999999999988e-23 < a < 1.89999999999999997e-33

   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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 86.8%

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

   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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.8

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

   7. Applied rewrites 85.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23} \lor \neg \left(a \leq 1.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift--.f64 N/A

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

   10. difference-of-squares N/A

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

   11. lower-/.f64 N/A

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

   12. difference-of-squares N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 86.5

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 86.5

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

6. Applied rewrites 86.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. *-inverses N/A

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

   10. *-rgt-identity N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 98.8

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 98.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.8

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

8. Applied rewrites 98.8%

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

Alternative 4: 63.1% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.9%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 55.6

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

7. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 61.8

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

9. Applied rewrites 61.8%

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

Alternative 5: 57.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.9%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 55.6

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

7. Applied rewrites 55.6%

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

Alternative 6: 57.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 86.9%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 55.6

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

7. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 55.6

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

9. Applied rewrites 55.6%

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

Reproduce

herbie shell --seed 2025026 
(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))))