NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 7.0s

Alternatives: 9

Speedup: 2.0×

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: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 83.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
2. Add Preprocessing
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. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 90.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

6. Applied rewrites 86.0%

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

8. Applied rewrites 99.7%

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

Alternative 2: 86.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15} \lor \neg \left(b \leq 3.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot a}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -4.5e+15) (not (<= b 3.1e+18)))
   (/ (/ (PI) b) (* 2.0 (* a b)))
   (/ (/ (* 0.5 (PI)) (* b a)) a)))

Derivation

1. Split input into 2 regimes

2. if b < -4.5e15 or 3.1e18 < b

   1. Initial program 82.0%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 95.2

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

   7. Applied rewrites 95.2%

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

   if -4.5e15 < b < 3.1e18

   1. Initial program 85.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.3%

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

   9. Applied rewrites 88.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15} \lor \neg \left(b \leq 3.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot a}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+15} \lor \neg \left(b \leq 3.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot \frac{0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -3.5e+15) (not (<= b 3.1e+18)))
   (* (/ (PI) (* (* b b) a)) 0.5)
   (* (/ (PI) (* b a)) (/ 0.5 a))))

Derivation

1. Split input into 2 regimes

2. if b < -3.5e15 or 3.1e18 < b

   1. Initial program 82.0%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*r* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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 88.7

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

   7. Applied rewrites 88.7%

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

   if -3.5e15 < b < 3.1e18

   1. Initial program 85.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.3%

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

   9. Applied rewrites 88.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+15} \lor \neg \left(b \leq 3.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot \frac{0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot b}}{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.5e+15)
   (* (/ (/ (PI) (* b b)) a) 0.5)
   (if (<= b 3.1e+18)
     (/ (/ (* 0.5 (PI)) (* b a)) a)
     (* (/ (PI) (* (* b b) a)) 0.5))))

Derivation

1. Split input into 3 regimes

2. if b < -4.5e15

   1. Initial program 83.8%

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

   3. 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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -4.5e15 < b < 3.1e18

   1. Initial program 85.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.3%

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

   9. Applied rewrites 88.9%

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

   if 3.1e18 < b

   1. Initial program 80.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*r* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot b}}{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot b}}{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.5e+15)
   (* (/ (/ (PI) (* b b)) a) 0.5)
   (if (<= b 3.1e+18)
     (/ (* (/ (PI) a) 0.5) (* b a))
     (* (/ (PI) (* (* b b) a)) 0.5))))

Derivation

1. Split input into 3 regimes

2. if b < -4.5e15

   1. Initial program 83.8%

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

   3. 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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -4.5e15 < b < 3.1e18

   1. Initial program 85.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 88.8%

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

   if 3.1e18 < b

   1. Initial program 80.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*r* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot b}}{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot b}}{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.5e+15)
   (* (/ (/ (PI) (* b b)) a) 0.5)
   (if (<= b 3.1e+18)
     (* (/ (PI) (* b a)) (/ 0.5 a))
     (* (/ (PI) (* (* b b) a)) 0.5))))

Derivation

1. Split input into 3 regimes

2. if b < -4.5e15

   1. Initial program 83.8%

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

   3. 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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if -4.5e15 < b < 3.1e18

   1. Initial program 85.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.3%

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

   9. Applied rewrites 88.8%

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

   if 3.1e18 < b

   1. Initial program 80.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*r* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot b}}{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+15} \lor \neg \left(b \leq 3.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(b \cdot a\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -3.5e+15) (not (<= b 3.1e+18)))
   (* (/ (PI) (* (* b b) a)) 0.5)
   (/ (* (PI) 0.5) (* (* b a) a))))

Derivation

1. Split input into 2 regimes

2. if b < -3.5e15 or 3.1e18 < b

   1. Initial program 82.0%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-rgt-identity N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*r* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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 88.7

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

   7. Applied rewrites 88.7%

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

   if -3.5e15 < b < 3.1e18

   1. Initial program 85.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 77.3%

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

   9. Applied rewrites 88.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+15} \lor \neg \left(b \leq 3.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(b \cdot a\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 83.6%

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

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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 60.9%

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

9. Applied rewrites 66.7%

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

10. Final simplification 66.7%

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

Alternative 9: 62.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 83.6%

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

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. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 60.9

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

5. Applied rewrites 60.9%

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

7. Applied rewrites 60.9%

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

9. Applied rewrites 66.7%

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

10. Final simplification 66.7%

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

Reproduce

herbie shell --seed 2024354 
(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))))