NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.6% → 99.7%

Time: 8.0s

Alternatives: 8

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: 78.6% 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: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

4. Applied rewrites 87.2%

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

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. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

4. Applied rewrites 87.2%

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

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. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-PI.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-*.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

4. Applied rewrites 87.2%

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

6. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-/.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-rgt-identity N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 99.7

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 4: 83.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 6e-118)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if b < 6.00000000000000035e-118

   1. Initial program 76.1%

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

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 72.5%

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

   if 6.00000000000000035e-118 < b

   1. Initial program 76.0%

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

   3. Taylor expanded in a around 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 72.6

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

   5. Applied rewrites 72.6%

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

Alternative 5: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

4. Applied rewrites 87.2%

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

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. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. associate-/r* N/A

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

   13. lift-PI.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.8

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lift-+.f64 98.8

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 98.8

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

10. Applied rewrites 98.8%

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

Alternative 6: 62.6% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 61.9%

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

Alternative 7: 56.6% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 55.0

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

5. Applied rewrites 55.0%

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

Alternative 8: 56.6% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 54.9%

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

Reproduce

herbie shell --seed 2024298 
(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))))