NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 7.6s

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: 78.2% 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, 2.0× speedup

Math

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

FPCore

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

Derivation

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 90.7%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+112)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ 0.5 (* (* (+ b a) a) b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999993e111

   1. Initial program 44.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 98.7%

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

   if -9.9999999999999993e111 < a

   1. Initial program 83.6%

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

   4. Applied rewrites 92.8%

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

   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. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. times-frac N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 98.1

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 96.4

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lift-+.f64 96.4

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

   10. Applied rewrites 96.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

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 90.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \frac{0.5}{b + a}}{b \cdot a} \]
10. 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 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 90.7%

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

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. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 99.6

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

10. Applied rewrites 99.6%

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

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: 71.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.15e-47)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if b < 2.1499999999999999e-47

   1. Initial program 81.4%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 64.8%

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

   9. Applied rewrites 71.4%

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

   if 2.1499999999999999e-47 < b

   1. Initial program 69.5%

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

   3. Taylor expanded in a around 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 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

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 90.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

   8. associate-/r/ N/A

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

   9. clear-num N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. clear-num N/A

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

   14. associate-/r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 98.2

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 98.2

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 98.2

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

8. Applied rewrites 98.2%

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

9. Final simplification 98.2%

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

Alternative 7: 62.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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 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 56.0

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

5. Applied rewrites 56.0%

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

7. Applied rewrites 60.8%

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

8. Final simplification 60.8%

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

Alternative 8: 62.3% 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 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 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 56.0

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

5. Applied rewrites 56.0%

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

7. Applied rewrites 56.0%

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

9. Applied rewrites 60.8%

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

10. Final simplification 60.8%

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

Alternative 9: 56.6% 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 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 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 56.0

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

5. Applied rewrites 56.0%

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

7. Applied rewrites 56.0%

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

8. Final simplification 56.0%

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

Reproduce

herbie shell --seed 2024308 
(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))))