NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.7%

Time: 8.2s

Alternatives: 9

Speedup: 2.4×

Specification

Math

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

FPCore

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.6%

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

4. Applied rewrites 88.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+121)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ 0.5 (* (* (+ a b) a) b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000007e121

   1. Initial program 62.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 99.8%

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

   if -5.00000000000000007e121 < a

   1. Initial program 79.1%

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

   4. Applied rewrites 90.9%

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

   6. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. distribute-rgt-out N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lower-*.f64 95.4

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lift-+.f64 95.4

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

   10. Applied rewrites 95.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.6%

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

4. Applied rewrites 88.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.6%

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

4. Applied rewrites 88.7%

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-rgt-identity N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.6

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 69.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e-30)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999989e-30

   1. Initial program 76.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 94.3%

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

   if -2.89999999999999989e-30 < a

   1. Initial program 76.7%

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

   3. 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 60.9

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

   5. Applied rewrites 60.9%

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

Alternative 6: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.6%

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

4. Applied rewrites 88.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

   8. div-inv N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 99.7

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 99.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. lower-*.f64 N/A

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 7: 62.6% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in 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 58.8

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

5. Applied rewrites 58.8%

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

7. Applied rewrites 65.7%

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

Alternative 8: 62.5% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in 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 58.8

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

5. Applied rewrites 58.8%

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

7. Applied rewrites 58.8%

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

9. Applied rewrites 65.7%

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

10. Final simplification 65.7%

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

Alternative 9: 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 76.6%

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

3. Taylor expanded in 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 58.8

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

5. Applied rewrites 58.8%

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

7. Applied rewrites 58.8%

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

8. Final simplification 58.8%

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

Reproduce

herbie shell --seed 2024295 
(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))))