NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 10.6s

Alternatives: 13

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: 78.1% 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, 1.7× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ (/ (PI) (* b 2.0)) a) (+ b a)))

Derivation

1. Initial program 80.5%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. div-inv N/A

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

   10. associate-*r* N/A

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

   11. div-inv N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r/ N/A

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

   8. lower-/.f64 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.4e+152)
   (* (PI) (/ 0.5 (* a (* b a))))
   (/ (* (PI) 0.5) (* b (* a (+ b a))))))

Derivation

1. Split input into 2 regimes

2. if a < -1.4000000000000001e152

   1. Initial program 54.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 98.5

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.9

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

   7. Applied rewrites 78.9%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 98.5

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

   9. Applied rewrites 98.5%

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

   if -1.4000000000000001e152 < a

   1. Initial program 84.3%

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*r* N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-rgt-in N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. distribute-rgt-in N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.5\\ \mathbf{if}\;b \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{t\_0}{a \cdot \left(b \cdot \left(b + a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (PI) 0.5)))
   (if (<= b 9.5e+81) (/ t_0 (* a (* b (+ b a)))) (/ t_0 (* b (* b a))))))

Derivation

1. Split input into 2 regimes

2. if b < 9.50000000000000083e81

   1. Initial program 81.8%

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*r* N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r/ N/A

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

      8. lower-/.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   10. Applied rewrites 93.7%

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

   if 9.50000000000000083e81 < b

   1. Initial program 72.7%

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

      1. lift-PI.f64 N/A

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

      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. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-sub N/A

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

      12. frac-times N/A

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

      13. 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 88.9%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.9

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

   9. Applied rewrites 98.9%

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

Alternative 4: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e+116)
   (/ (* (PI) 0.5) (* a (* b a)))
   (* (PI) (/ 0.5 (* b (* a (+ b a)))))))

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000002e116

   1. Initial program 62.5%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.00000000000000002e116 < a

   1. Initial program 83.8%

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*r* N/A

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

      11. div-inv N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-in N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-in N/A

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

      20. +-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. lower-*.f64 96.6

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

   8. Applied rewrites 96.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{b \cdot \left(a \cdot \left(b + a\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (/ (* (PI) 0.5) (* b a)) (+ b a)))

Derivation

1. Initial program 80.5%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. div-inv N/A

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

   10. associate-*r* N/A

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

   11. div-inv N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ (PI) (+ b a)) (/ 0.5 (* b a))))

Derivation

1. Initial program 80.5%

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ (PI) (* b a)) (/ 0.5 (+ b a))))

Derivation

1. Initial program 80.5%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. div-inv N/A

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

   10. associate-*r* N/A

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

   11. div-inv N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. associate-*l/ N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 8: 89.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.5\\ \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{t\_0}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (PI) 0.5)))
   (if (<= b 1.1e-95) (/ t_0 (* a (* b a))) (/ t_0 (* b (* b a))))))

Derivation

1. Split input into 2 regimes

2. if b < 1.0999999999999999e-95

   1. Initial program 78.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if 1.0999999999999999e-95 < b

   1. Initial program 85.9%

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

      1. lift-PI.f64 N/A

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

      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. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-sub N/A

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

      12. frac-times N/A

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

      13. 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 94.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 78.3

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

   7. Applied rewrites 78.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.8

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

   9. Applied rewrites 77.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-95)
   (/ (* (PI) 0.5) (* a (* b a)))
   (* (PI) (/ 0.5 (* b (* b a))))))

Derivation

1. Split input into 2 regimes

2. if b < 1.0999999999999999e-95

   1. Initial program 78.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if 1.0999999999999999e-95 < b

   1. Initial program 85.9%

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

      1. lift-PI.f64 N/A

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

      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. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-sub N/A

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

      12. frac-times N/A

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

      13. 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 94.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 78.3

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

   7. Applied rewrites 78.3%

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

      1. lift-PI.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-/l* N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. div-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l/ N/A

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

      15. associate-/l/ N/A

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

      16. associate-/r* N/A

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

      17. metadata-eval N/A

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

      18. associate-/l/ N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.7

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

   9. Applied rewrites 77.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-95)
   (* (PI) (/ 0.5 (* a (* b a))))
   (* (PI) (/ 0.5 (* b (* b a))))))

Derivation

1. Split input into 2 regimes

2. if b < 1.0999999999999999e-95

   1. Initial program 78.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 67.8

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

   9. Applied rewrites 67.8%

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

   if 1.0999999999999999e-95 < b

   1. Initial program 85.9%

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

      1. lift-PI.f64 N/A

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

      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. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-sub N/A

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

      12. frac-times N/A

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

      13. 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 94.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 78.3

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

   7. Applied rewrites 78.3%

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

      1. lift-PI.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-/l* N/A

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

      8. div-inv N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. div-inv N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l/ N/A

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

      15. associate-/l/ N/A

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

      16. associate-/r* N/A

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

      17. metadata-eval N/A

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

      18. associate-/l/ N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.7

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

   9. Applied rewrites 77.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{a \cdot \left(b \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 2.4× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ (* (PI) 0.5) (* (+ b a) (* b a))))

Derivation

1. Initial program 80.5%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. clear-num N/A

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

   14. frac-times N/A

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

   15. *-rgt-identity N/A

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

6. Applied rewrites 99.2%

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

Alternative 12: 63.6% accurate, 2.6× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (PI) (/ 0.5 (* a (* b a)))))

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 64.0

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

5. Applied rewrites 64.0%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 63.9

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 59.2

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

7. Applied rewrites 59.2%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 63.9

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

9. Applied rewrites 63.9%

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

10. Final simplification 63.9%

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

Alternative 13: 57.6% accurate, 2.6× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (PI) (/ 0.5 (* b (* a a)))))

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 64.0

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

5. Applied rewrites 64.0%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 63.9

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 59.2

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

7. Applied rewrites 59.2%

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

Reproduce

herbie shell --seed 2024216 
(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))))