NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 98.8%

Time: 9.7s

Alternatives: 20

Speedup: 2.2×

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: 78.4% 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: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(\frac{b - a}{a} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{0.5}{b - a}}{\left(a + b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e+77)
   (/ (* (* (/ (- b a) a) (PI)) (/ 0.5 (- b a))) (* (+ a b) b))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if b < 2.50000000000000002e77

   1. Initial program 78.7%

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

   4. Applied rewrites 84.2%

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

   6. Applied rewrites 90.1%

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

   if 2.50000000000000002e77 < b

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(\frac{b - a}{a} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{0.5}{b - a}}{\left(a + b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 83.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 98.7

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

6. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(\frac{b - a}{a} \cdot \mathsf{PI}\left(\right)\right) \cdot 0.5}{\left(\left(a + b\right) \cdot b\right) \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e+77)
   (/ (* (* (/ (- b a) a) (PI)) 0.5) (* (* (+ a b) b) (- b a)))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if b < 2.50000000000000002e77

   1. Initial program 78.7%

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

   4. Applied rewrites 84.2%

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

   6. Applied rewrites 85.4%

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

   if 2.50000000000000002e77 < b

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 4: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5}{\frac{\left(\left(a + b\right) \cdot b\right) \cdot a}{\mathsf{PI}\left(\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e+77)
   (/ 0.5 (* (/ (* (* (+ a b) b) a) (* (PI) (- b a))) (- b a)))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if b < 2.8e77

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.9

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 94.7%

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

   if 2.8e77 < b

   1. Initial program 59.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5}{\frac{\left(\left(a + b\right) \cdot b\right) \cdot a}{\mathsf{PI}\left(\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 83.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 98.7

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

6. Applied rewrites 98.7%

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

7. Final simplification 98.7%

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

Alternative 6: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-49}:\\ \;\;\;\;\left(\frac{\mathsf{PI}\left(\right)}{a + b} \cdot \frac{-1}{b}\right) \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.42e-49)
   (* (* (/ (PI) (+ a b)) (/ -1.0 b)) (/ 0.5 (- b a)))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if a < -1.42e-49

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto \left(\frac{\color{blue}{-1}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{a + b}\right) \cdot \frac{\frac{1}{2}}{b - a} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.7%

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

   if -1.42e-49 < a

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 77.0

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

   7. Applied rewrites 77.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-49}:\\ \;\;\;\;\left(\frac{\mathsf{PI}\left(\right)}{a + b} \cdot \frac{-1}{b}\right) \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{-\mathsf{PI}\left(\right)}{a}}{b} \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.42e-49)
   (* (/ (/ (- (PI)) a) b) (/ 0.5 (- b a)))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if a < -1.42e-49

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)} \cdot \frac{\frac{1}{2}}{b - a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-PI.f64 92.6

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

   7. Applied rewrites 92.6%

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

   if -1.42e-49 < a

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 77.0

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

   7. Applied rewrites 77.0%

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

Alternative 8: 91.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.42 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.42e-49)
   (* (/ (- (PI)) (* a b)) (/ 0.5 (- b a)))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if a < -1.42e-49

   1. Initial program 70.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot b}} \cdot \frac{\frac{1}{2}}{b - a} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 51.7

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

   7. Applied rewrites 51.7%

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

   8. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\right)} \cdot \frac{\frac{1}{2}}{b - a} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 92.5

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

   10. Applied rewrites 92.5%

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

   if -1.42e-49 < a

   1. Initial program 75.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 77.0

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

   7. Applied rewrites 77.0%

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

Alternative 9: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.92 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{\frac{0.5}{b - a}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.92e-147)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (* (PI) (/ (/ 0.5 (- b a)) (* a b)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.9200000000000001e-147

   1. Initial program 75.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 57.5

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 71.2%

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

   if 1.9200000000000001e-147 < b

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 98.6

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

   6. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{\frac{\frac{1}{2}}{b - a}}{a \cdot b} \]
   10. Step-by-step derivation

       1. lower-PI.f64 90.4

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

   11. Applied rewrites 90.4%

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

Alternative 10: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.92 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.92e-147)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (* (/ (PI) (* a b)) (/ 0.5 (- b a)))))

Derivation

1. Split input into 2 regimes

2. if b < 1.9200000000000001e-147

   1. Initial program 75.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 57.5

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 71.2%

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

   if 1.9200000000000001e-147 < b

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot b}} \cdot \frac{\frac{1}{2}}{b - a} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{a \cdot b}} \cdot \frac{\frac{1}{2}}{b - a} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 90.3

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

   10. Applied rewrites 90.3%

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

Alternative 11: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (/ (* (/ (PI) a) 0.5) (* a b))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if -6e-15 < a

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{b}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 77.4

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

   7. Applied rewrites 77.4%

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

Alternative 12: 83.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\frac{b}{\mathsf{PI}\left(\right)} \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (/ (* (/ (PI) a) 0.5) (* a b))
   (/ 0.5 (* (* (/ b (PI)) b) a))))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a}}}{a \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if -6e-15 < a

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a \cdot {b}^{2}}{\mathsf{PI}\left(\right)}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-PI.f64 64.1

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

   11. Applied rewrites 64.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\frac{b}{\mathsf{PI}\left(\right)} \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\frac{b}{\mathsf{PI}\left(\right)} \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (/ 0.5 (* (* (/ b (PI)) b) a))))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 85.4%

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

   11. Applied rewrites 87.0%

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

   if -6e-15 < a

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{a \cdot {b}^{2}}{\mathsf{PI}\left(\right)}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-PI.f64 64.1

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

   11. Applied rewrites 64.1%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\frac{b}{\mathsf{PI}\left(\right)} \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 85.4%

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

   11. Applied rewrites 87.0%

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

   if -6e-15 < a

   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 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 63.7

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

   5. Applied rewrites 63.7%

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

Alternative 15: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (* (/ 0.5 (* a b)) (/ (PI) a))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 87.0%

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

   if -6e-15 < a

   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 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 63.7

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

   5. Applied rewrites 63.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 83.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (* (/ (/ 0.5 a) (* a b)) (PI))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.7%

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

   9. Applied rewrites 86.9%

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

   if -6e-15 < a

   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 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 63.7

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

   5. Applied rewrites 63.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -6e-15)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6e-15

   1. Initial program 68.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 85.4%

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

   if -6e-15 < a

   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 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 63.7

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

   5. Applied rewrites 63.7%

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

Alternative 18: 64.1% 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 74.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 49.3

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

5. Applied rewrites 49.3%

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

7. Applied rewrites 58.0%

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

Alternative 19: 64.1% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 49.3

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

5. Applied rewrites 49.3%

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

7. Applied rewrites 49.2%

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

9. Applied rewrites 58.0%

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

10. Final simplification 58.0%

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

Alternative 20: 57.6% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 49.3

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

5. Applied rewrites 49.3%

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

7. Applied rewrites 49.2%

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

8. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2024327 
(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))))