NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.0%

Time: 4.7s

Alternatives: 7

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: 78.9% 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: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.16 \cdot 10^{+168} \lor \neg \left(a \leq 6.5 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(-a\right)} \cdot \frac{-1}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.16e+168) (not (<= a 6.5e+95)))
   (* (/ (/ (PI) a) (* b a)) 0.5)
   (* (/ (PI) (* (* (+ b a) 2.0) (- a))) (/ -1.0 b))))

Derivation

1. Split input into 2 regimes

2. if a < -1.1599999999999999e168 or 6.5e95 < a

   1. Initial program 65.9%

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

   3. 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. lower-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. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 97.9

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

   7. Applied rewrites 97.9%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -1.1599999999999999e168 < a < 6.5e95

   1. Initial program 85.9%

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

      1. frac-times N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 53.1

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

   7. Applied rewrites 53.1%

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

   8. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.6

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

   10. Applied rewrites 99.6%

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

       1. *-rgt-identity N/A

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

       2. lower-/.f64 N/A

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

       3. lower-PI.f64 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. lower-neg.f64 99.6

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

   12. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(-a\right)}} \cdot \frac{-1}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.16 \cdot 10^{+168} \lor \neg \left(a \leq 6.5 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(\left(b + a\right) \cdot 2\right) \cdot \left(-a\right)} \cdot \frac{-1}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot {\left(b \cdot a\right)}^{-1}}{b + a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (* (/ (PI) 2.0) (pow (* b a) -1.0)) (+ b a)))

Derivation

1. Initial program 79.6%

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

   1. associate-*r/ N/A

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

   2. difference-of-squares N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.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. lower-PI.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 88.5

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

4. Applied rewrites 88.5%

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

   1. associate-*l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. inv-pow N/A

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

   14. lower-pow.f64 N/A

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

   15. inv-pow N/A

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

   16. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

    1. associate-*l/ N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-PI.f64 N/A

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

    6. lower-pow.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot {\left(b \cdot a\right)}^{-1}}{\mathsf{Rewrite<=}\left(+-commutative, \left(b + a\right)\right)} \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{2} \cdot {\left(b \cdot a\right)}^{-1}}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(b + a\right)\right)} \]

11. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{PI}\left(\right)}{2}}{a + b} \cdot \frac{1}{b \cdot a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ (/ (PI) 2.0) (+ a b)) (/ 1.0 (* b a))))

Derivation

1. Initial program 79.6%

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

   1. associate-*r/ N/A

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

   2. difference-of-squares N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.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. lower-PI.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 88.5

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

4. Applied rewrites 88.5%

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

   1. associate-*l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. inv-pow N/A

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

   14. lower-pow.f64 N/A

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

   15. inv-pow N/A

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

   16. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.6

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

9. Applied rewrites 99.6%

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

    1. unpow-1 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 4: 86.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -34000 \lor \neg \left(a \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -34000.0) (not (<= a 3.4e-17)))
   (* (/ (/ (PI) a) (* b a)) 0.5)
   (* (/ (PI) (* b (* b a))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -34000 or 3.3999999999999998e-17 < a

   1. Initial program 78.0%

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

   3. Taylor expanded in a around 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. lower-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. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 93.1

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

   9. Applied rewrites 93.1%

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

   if -34000 < a < 3.3999999999999998e-17

   1. Initial program 81.4%

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

      1. associate-*r/ N/A

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

      2. difference-of-squares N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.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. lower-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-pow.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   8. 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. lower-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. lower-*.f64 72.0

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

   9. Applied rewrites 72.0%

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

       1. associate-*l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 86.6

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

   11. Applied rewrites 86.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -34000 \lor \neg \left(a \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -34000 \lor \neg \left(a \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -34000.0) (not (<= a 3.4e-17)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* b (* b a))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -34000 or 3.3999999999999998e-17 < a

   1. Initial program 78.0%

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

   3. Taylor expanded in a around 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. lower-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. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

   if -34000 < a < 3.3999999999999998e-17

   1. Initial program 81.4%

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

      1. associate-*r/ N/A

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

      2. difference-of-squares N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.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. lower-PI.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 85.5

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

   4. Applied rewrites 85.5%

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-pow.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   8. 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. lower-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. lower-*.f64 72.0

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

   9. Applied rewrites 72.0%

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

       1. associate-*l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 86.6

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

   11. Applied rewrites 86.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -34000 \lor \neg \left(a \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -34000 \lor \neg \left(a \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -34000.0) (not (<= a 3.4e-17)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -34000 or 3.3999999999999998e-17 < a

   1. Initial program 78.0%

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

   3. Taylor expanded in a around 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. lower-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. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. associate-*l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 91.7

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

   7. Applied rewrites 91.7%

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

   if -34000 < a < 3.3999999999999998e-17

   1. Initial program 81.4%

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

   3. 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. lower-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. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -34000 \lor \neg \left(a \leq 3.4 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% 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 79.6%

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

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

   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. lower-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. lower-*.f64 59.7

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

5. Applied rewrites 59.7%

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

   1. associate-*l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 64.0

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

7. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025044 
(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))))