NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.2% → 99.7%

Time: 8.1s

Alternatives: 7

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. *-rgt-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. *-lft-identity N/A

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

   21. *-rgt-identity N/A

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

4. Applied rewrites 88.0%

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

6. Applied rewrites 81.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

8. Applied rewrites 99.7%

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

Alternative 2: 86.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b}\\ t_1 := \frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -1020000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -0.2:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{-7} \lor \neg \left(a \leq 1.9 \cdot 10^{+284}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (PI) (/ 0.5 (* (* a b) b))))
        (t_1 (/ (* 0.5 (PI)) (* (* a b) a))))
   (if (<= a -7.8e+31)
     t_1
     (if (<= a -1020000000.0)
       t_0
       (if (<= a -0.2)
         (* (PI) (/ 0.5 (* (* a a) b)))
         (if (or (<= a 9.6e-7) (not (<= a 1.9e+284))) t_0 t_1))))))

Derivation

1. Split input into 3 regimes

2. if a < -7.79999999999999999e31 or 9.59999999999999914e-7 < a < 1.8999999999999999e284

   1. Initial program 79.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 81.4

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

   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 81.3%

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

   9. Applied rewrites 92.7%

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

   if -7.79999999999999999e31 < a < -1.02e9 or -0.20000000000000001 < a < 9.59999999999999914e-7 or 1.8999999999999999e284 < a

   1. Initial program 82.4%

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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. associate-*r* N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. *-rgt-identity N/A

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 70.5%

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

   11. Applied rewrites 82.6%

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

   if -1.02e9 < a < -0.20000000000000001

   1. Initial program 100.0%

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

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq -1020000000:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b}\\ \mathbf{elif}\;a \leq -0.2:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot a\right) \cdot b}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{-7} \lor \neg \left(a \leq 1.9 \cdot 10^{+284}\right):\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000375 \lor \neg \left(b \leq -2.6 \cdot 10^{-300} \lor \neg \left(b \leq -7.2 \cdot 10^{-301} \lor \neg \left(b \leq 1.52 \cdot 10^{-37}\right)\right)\right):\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -0.000375)
         (not
          (or (<= b -2.6e-300)
              (not (or (<= b -7.2e-301) (not (<= b 1.52e-37)))))))
   (* (PI) (/ 0.5 (* (* a b) b)))
   (* (PI) (/ 0.5 (* (* a a) b)))))

Derivation

1. Split input into 2 regimes

2. if b < -3.7500000000000001e-4 or -2.59999999999999997e-300 < b < -7.20000000000000015e-301 or 1.52e-37 < b

   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. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. associate-*r* N/A

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

      11. *-lft-identity N/A

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

      12. *-rgt-identity N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. *-rgt-identity N/A

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

   9. Applied rewrites 76.5%

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

   11. Applied rewrites 88.1%

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

   if -3.7500000000000001e-4 < b < -2.59999999999999997e-300 or -7.20000000000000015e-301 < b < 1.52e-37

   1. Initial program 82.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.4%

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

   9. Applied rewrites 74.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000375 \lor \neg \left(b \leq -2.6 \cdot 10^{-300} \lor \neg \left(b \leq -7.2 \cdot 10^{-301} \lor \neg \left(b \leq 1.52 \cdot 10^{-37}\right)\right)\right):\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. *-rgt-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. *-lft-identity N/A

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

   21. *-rgt-identity N/A

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

4. Applied rewrites 88.0%

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

6. Applied rewrites 81.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 99.1

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.1

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

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

Alternative 5: 93.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. *-rgt-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. *-lft-identity N/A

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

   21. *-rgt-identity N/A

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

4. Applied rewrites 88.0%

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

6. Applied rewrites 81.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 92.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 92.7

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

10. Applied rewrites 92.7%

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

Alternative 6: 63.1% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. *-rgt-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. *-lft-identity N/A

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

   21. *-rgt-identity N/A

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 53.8

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

7. Applied rewrites 53.8%

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

9. Applied rewrites 60.3%

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

Alternative 7: 63.1% 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 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. difference-of-squares N/A

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

   10. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. *-rgt-identity N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. *-lft-identity N/A

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

   21. *-rgt-identity N/A

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 53.8

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

7. Applied rewrites 53.8%

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

9. Applied rewrites 53.8%

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

11. Applied rewrites 60.3%

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

Reproduce

herbie shell --seed 2024338 
(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))))