NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.4% → 99.4%

Time: 8.4s

Alternatives: 9

Speedup: 2.2×

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: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6.2e+95)
   (* (/ (PI) a) (/ 0.5 (* a b)))
   (/ (* (PI) 0.5) (* b (* a (+ a b))))))

Derivation

1. Split input into 2 regimes

2. if a < -6.2000000000000006e95

   1. Initial program 54.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 99.9%

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

   if -6.2000000000000006e95 < a

   1. Initial program 80.7%

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

      1. lift-*.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. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. difference-of-squares N/A

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

      12. lift--.f64 N/A

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

      13. flip-+ N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. div-inv N/A

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

      19. associate-/r* N/A

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

      20. *-commutative N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 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. lower-+.f64 94.5

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

   8. Applied rewrites 94.5%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{b \cdot \left(a \cdot \left(a + b\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.7e+92)
   (* (PI) (/ 0.5 (* (* b a) a)))
   (/ (* (PI) 0.5) (* b (* a (+ a b))))))

Derivation

1. Split input into 2 regimes

2. if a < -1.6999999999999999e92

   1. Initial program 55.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 96.3%

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

   if -1.6999999999999999e92 < a

   1. Initial program 80.6%

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

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. difference-of-squares N/A

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

      12. lift--.f64 N/A

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

      13. flip-+ N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. div-inv N/A

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

      19. associate-/r* N/A

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

      20. *-commutative N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 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. lower-+.f64 94.5

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

   8. Applied rewrites 94.5%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-sub N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. frac-times N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 83.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. +-commutative N/A

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

   11. difference-of-squares N/A

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

   12. lift--.f64 N/A

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

   13. flip-+ N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. associate-/r* N/A

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

   20. *-commutative N/A

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

6. Applied rewrites 99.7%

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

Alternative 4: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-sub N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. frac-times N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 83.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. +-commutative N/A

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

   11. difference-of-squares N/A

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

   12. lift--.f64 N/A

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

   13. flip-+ N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. associate-/r* N/A

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

   20. *-commutative N/A

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r/ N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-+.f64 99.1

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

8. Applied rewrites 99.1%

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

Alternative 5: 89.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.12e-33)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* a b) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.11999999999999999e-33

   1. Initial program 69.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 92.2%

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

   if -1.11999999999999999e-33 < a

   1. Initial program 78.1%

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

   4. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 66.7

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 72.9%

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

Alternative 6: 83.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e-33)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.10000000000000003e-33

   1. Initial program 69.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 92.2%

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

   if -1.10000000000000003e-33 < a

   1. Initial program 78.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

Alternative 7: 63.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 59.3

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

5. Applied rewrites 59.3%

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

7. Applied rewrites 66.8%

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

Alternative 8: 63.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 59.3

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

5. Applied rewrites 59.3%

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

7. Applied rewrites 59.3%

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

9. Applied rewrites 66.8%

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

Alternative 9: 57.9% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 59.3

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

5. Applied rewrites 59.3%

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

7. Applied rewrites 59.3%

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

Reproduce

herbie shell --seed 2024313 
(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))))