NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.4%

Time: 4.5s

Alternatives: 9

Speedup: 1.8×

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: 78.3% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+57} \lor \neg \left(a \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{2 \cdot \left(a + b\right)} \cdot \left({\left(b - a\right)}^{-1} \cdot \frac{\frac{b - a}{a}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot 2} \cdot \frac{\frac{b - a}{b}}{a}}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.2e+57) (not (<= a 5e+15)))
   (* (/ (PI) (* 2.0 (+ a b))) (* (pow (- b a) -1.0) (/ (/ (- b a) a) b)))
   (/ (* (/ (PI) (* (+ a b) 2.0)) (/ (/ (- b a) b) a)) (- b a))))

Derivation

1. Split input into 2 regimes

2. if a < -1.20000000000000002e57 or 5e15 < a

   1. Initial program 78.1%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 92.2

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

   4. Applied rewrites 92.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.8%

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

   if -1.20000000000000002e57 < a < 5e15

   1. Initial program 85.4%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 91.2

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

   4. Applied rewrites 91.2%

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

   6. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

   8. Applied rewrites 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+57} \lor \neg \left(a \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{2 \cdot \left(a + b\right)} \cdot \left({\left(b - a\right)}^{-1} \cdot \frac{\frac{b - a}{a}}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot 2} \cdot \frac{\frac{b - a}{b}}{a}}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3000000000000 \lor \neg \left(b \leq 2 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot 2} \cdot \frac{\frac{b - a}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{a}}{\left(b \cdot \left(\left(b + a\right) \cdot 2\right)\right) \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -3000000000000.0) (not (<= b 2e-42)))
   (/ (* (/ (PI) (* (+ a b) 2.0)) (/ (/ (- b a) b) a)) (- b a))
   (/ (/ (* (- b a) (PI)) a) (* (* b (* (+ b a) 2.0)) (- b a)))))

Derivation

1. Split input into 2 regimes

2. if b < -3e12 or 2.00000000000000008e-42 < b

   1. Initial program 79.1%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 90.5

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

   4. Applied rewrites 90.5%

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

   6. Applied rewrites 90.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-lft-identity N/A

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

      13. *-rgt-identity N/A

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

   8. Applied rewrites 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   10. Applied rewrites 99.7%

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

   if -3e12 < b < 2.00000000000000008e-42

   1. Initial program 86.1%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 92.9

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

   4. Applied rewrites 92.9%

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

   6. Applied rewrites 93.0%

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

   8. Applied rewrites 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3000000000000 \lor \neg \left(b \leq 2 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot 2} \cdot \frac{\frac{b - a}{b}}{a}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{a}}{\left(b \cdot \left(\left(b + a\right) \cdot 2\right)\right) \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+123} \lor \neg \left(b \leq 2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1e+123) (not (<= b 2e+107)))
   (/ (* (/ (PI) b) 0.5) (* a b))
   (/ (* (/ (PI) (* (* 2.0 (+ a b)) (- b a))) (- b a)) (* a b))))

Derivation

1. Split input into 2 regimes

2. if b < -9.99999999999999978e122 or 1.9999999999999999e107 < b

   1. Initial program 61.0%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 82.6

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 81.6%

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

   7. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -9.99999999999999978e122 < b < 1.9999999999999999e107

   1. Initial program 90.9%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 95.2

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

   4. Applied rewrites 95.2%

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

   6. Applied rewrites 95.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+123} \lor \neg \left(b \leq 2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{a}}{\left(b \cdot \left(\left(b + a\right) \cdot 2\right)\right) \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -9.8e+76)
   (/ (* (/ (PI) b) 0.5) (* a b))
   (if (<= b 3.1e+62)
     (/ (/ (* (- b a) (PI)) a) (* (* b (* (+ b a) 2.0)) (- b a)))
     (* (/ (PI) (* b (* a b))) 0.5))))

Derivation

1. Split input into 3 regimes

2. if b < -9.80000000000000053e76

   1. Initial program 71.5%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 86.3

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

   4. Applied rewrites 86.3%

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -9.80000000000000053e76 < b < 3.10000000000000014e62

   1. Initial program 89.6%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 94.6

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

   4. Applied rewrites 94.6%

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

   6. Applied rewrites 94.7%

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

   8. Applied rewrites 99.1%

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

   if 3.10000000000000014e62 < b

   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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 86.6

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

   4. Applied rewrites 86.6%

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

   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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 86.7

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

   7. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{a}}{\left(b \cdot \left(\left(b + a\right) \cdot 2\right)\right) \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+85} \lor \neg \left(b \leq 1.55 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -3.7e+85) (not (<= b 1.55e+118)))
   (/ (* (/ (PI) b) 0.5) (* a b))
   (/ (* (PI) b) (* (* a b) (* 2.0 (* (+ b a) b))))))

Derivation

1. Split input into 2 regimes

2. if b < -3.7000000000000002e85 or 1.54999999999999993e118 < b

   1. Initial program 64.9%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 84.3

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

   4. Applied rewrites 84.3%

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

   6. Applied rewrites 83.4%

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

   7. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -3.7000000000000002e85 < b < 1.54999999999999993e118

   1. Initial program 90.4%

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

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 65.1

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

   7. Applied rewrites 65.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+85} \lor \neg \left(b \leq 1.55 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-38} \lor \neg \left(b \leq 1.02 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.8e-38) (not (<= b 1.02e-41)))
   (* (/ (PI) (* b (* a b))) 0.5)
   (/ (* (/ (PI) a) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if b < -1.8e-38 or 1.02e-41 < b

   1. Initial program 81.2%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 91.4

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

   4. Applied rewrites 91.4%

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

   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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 85.4

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

   9. Applied rewrites 85.4%

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

   if -1.8e-38 < b < 1.02e-41

   1. Initial program 84.1%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 91.9

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

   4. Applied rewrites 91.9%

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 89.2

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

   9. Applied rewrites 89.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-38} \lor \neg \left(b \leq 1.02 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-38} \lor \neg \left(b \leq 1.02 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.8e-38) (not (<= b 1.02e-41)))
   (* (/ (PI) (* b (* a b))) 0.5)
   (* (/ (PI) (* a (* a b))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if b < -1.8e-38 or 1.02e-41 < b

   1. Initial program 81.2%

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

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. difference-of-squares N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 91.4

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

   4. Applied rewrites 91.4%

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

   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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 85.4

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

   9. Applied rewrites 85.4%

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

   if -1.8e-38 < b < 1.02e-41

   1. Initial program 84.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-38} \lor \neg \left(b \leq 1.02 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-38} \lor \neg \left(b \leq 1.02 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.8e-38) (not (<= b 1.02e-41)))
   (* (/ (PI) (* (* b b) a)) 0.5)
   (* (/ (PI) (* a (* a b))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if b < -1.8e-38 or 1.02e-41 < b

   1. Initial program 81.2%

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

   3. 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -1.8e-38 < b < 1.02e-41

   1. Initial program 84.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-38} \lor \neg \left(b \leq 1.02 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in a around 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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 58.5

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

5. Applied rewrites 58.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 61.4

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

7. Applied rewrites 61.4%

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

Reproduce

herbie shell --seed 2025051 
(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))))