NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.6%

Time: 4.7s

Alternatives: 7

Speedup: 2.0×

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: 78.6% 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: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 85.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{0.5}{b \cdot a}}{a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.3e+105)
   (* (/ (/ 0.5 (* b a)) a) (PI))
   (/ (* 0.5 (PI)) (* (* (+ b a) a) b))))

Derivation

1. Split input into 2 regimes

2. if a < -1.3000000000000001e105

   1. Initial program 71.4%

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

   3. 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 90.4

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 97.8%

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

   11. Applied rewrites 99.8%

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

   if -1.3000000000000001e105 < a

   1. Initial program 84.2%

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot a}{0.5 \cdot \mathsf{PI}\left(\right)} \cdot \left(b + a\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 95.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.5

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.5

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 95.5

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

   10. Applied rewrites 95.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{0.5}{b \cdot a}}{a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+102}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3e+102)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (/ (* 0.5 (PI)) (* (* (+ b a) a) b))))

Derivation

1. Split input into 2 regimes

2. if a < -2.9999999999999998e102

   1. Initial program 72.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 90.7%

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

   9. Applied rewrites 97.8%

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

   if -2.9999999999999998e102 < a

   1. Initial program 84.1%

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

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 83.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot a}{0.5 \cdot \mathsf{PI}\left(\right)} \cdot \left(b + a\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 95.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 95.5

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 95.5

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.5

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 95.5

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

   10. Applied rewrites 95.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+102}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 85.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot a}{0.5 \cdot \mathsf{PI}\left(\right)} \cdot \left(b + a\right)}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. lift-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.6

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.6

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

10. Applied rewrites 99.6%

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

11. Final simplification 99.6%

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

Alternative 5: 69.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e-44)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -5.8000000000000003e-44

   1. Initial program 83.0%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 82.7%

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

   9. Applied rewrites 87.0%

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

   if -5.8000000000000003e-44 < a

   1. Initial program 82.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 58.5%

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

9. Applied rewrites 62.2%

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

10. Final simplification 62.2%

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

Alternative 7: 56.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.5

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

5. Applied rewrites 58.5%

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

7. Applied rewrites 58.5%

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

8. Final simplification 58.5%

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

Reproduce

herbie shell --seed 2024332 
(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))))