NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.1%

Time: 6.0s

Alternatives: 5

Speedup: 2.4×

Specification

Math

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

FPCore

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 83.4%

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

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   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. *-rgt-identity N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

   17. lower-/.f64 N/A

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

   18. *-lft-identity N/A

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

   19. *-rgt-identity N/A

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

   20. lower--.f64 92.7

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

4. Applied rewrites 92.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 99.3%

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

Alternative 2: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.06 \cdot 10^{-64} \lor \neg \left(a \leq 56000000\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.06e-64) (not (<= a 56000000.0)))
   (/ (* (PI) 0.5) (* (* a a) b))
   (* (/ (PI) (* (* a b) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.06000000000000007e-64 or 5.6e7 < a

   1. Initial program 81.0%

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

   5. Applied rewrites 81.4%

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

   7. Applied rewrites 81.8%

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

   if -1.06000000000000007e-64 < a < 5.6e7

   1. Initial program 86.2%

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

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. *-rgt-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

      17. lower-/.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 92.6

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

   4. Applied rewrites 92.6%

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

   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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 87.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.06 \cdot 10^{-64} \lor \neg \left(a \leq 56000000\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(\left(a + b\right) \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.2e+132)
   (* (PI) (/ 0.5 (* (* a b) b)))
   (/ (* 0.5 (PI)) (* (* (+ a b) b) a))))

Derivation

1. Split input into 2 regimes

2. if b < -1.2000000000000001e132

   1. Initial program 59.8%

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

      1. lift-*.f64 N/A

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

      2. 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. associate-*r/ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. *-rgt-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

      17. lower-/.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 83.8

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

   4. Applied rewrites 83.8%

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

   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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 81.5%

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

   11. Applied rewrites 99.6%

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

   if -1.2000000000000001e132 < b

   1. Initial program 88.5%

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

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      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. *-rgt-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

      17. lower-/.f64 N/A

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

      18. *-lft-identity N/A

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

      19. *-rgt-identity N/A

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

      20. lower--.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 99.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 98.8

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-+.f64 98.8

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

   11. Applied rewrites 98.8%

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

Alternative 4: 62.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 83.4%

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

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   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. *-rgt-identity N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

   17. lower-/.f64 N/A

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

   18. *-lft-identity N/A

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

   19. *-rgt-identity N/A

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

   20. lower--.f64 92.7

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

4. Applied rewrites 92.7%

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

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

7. Applied rewrites 62.9%

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

9. Applied rewrites 66.6%

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

Alternative 5: 62.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 83.4%

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

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   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. *-rgt-identity N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

   17. lower-/.f64 N/A

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

   18. *-lft-identity N/A

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

   19. *-rgt-identity N/A

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

   20. lower--.f64 92.7

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

4. Applied rewrites 92.7%

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

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

7. Applied rewrites 62.9%

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

9. Applied rewrites 62.9%

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

11. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025019 
(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))))