NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.1% → 99.7%

Time: 8.0s

Alternatives: 7

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: 79.1% 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

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

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 78.6%

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

4. Applied rewrites 87.4%

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

6. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-/.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

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 78.6%

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

4. Applied rewrites 87.4%

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

6. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-/.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 3: 84.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e-69)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.8999999999999999e-69

   1. Initial program 70.7%

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

   3. Taylor expanded in a around 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 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 83.6%

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

   if -1.8999999999999999e-69 < a

   1. Initial program 81.6%

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

   3. Taylor expanded in a around 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 64.6

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

   5. Applied rewrites 64.6%

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

Alternative 4: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.6%

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

4. Applied rewrites 87.4%

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

6. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-/.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.5

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 99.5

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 5: 63.4% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.3

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

5. Applied rewrites 57.3%

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

7. Applied rewrites 65.3%

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

Alternative 6: 63.4% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 78.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.3

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

5. Applied rewrites 57.3%

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

7. Applied rewrites 57.3%

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

9. Applied rewrites 65.3%

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

Alternative 7: 58.3% 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 78.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.3

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

5. Applied rewrites 57.3%

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

7. Applied rewrites 57.3%

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

Reproduce

herbie shell --seed 2024320 
(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))))