NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.9% → 99.5%

Time: 7.4s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.5}{a + b} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{{b}^{-1}}{a} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ 0.5 (+ a b)) (PI)) (/ (pow b -1.0) a)))

Derivation

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. 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 87.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-sub N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. frac-times N/A

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

   12. *-commutative N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 92.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 92.6

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

8. Applied rewrites 92.6%

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

9. Taylor expanded in a around 0

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

    1. *-commutative N/A

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

    2. associate-/r* N/A

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

    3. lower-/.f64 N/A

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

    4. lower-/.f64 99.5

       \[\leadsto \left(\frac{0.5}{a + b} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\color{blue}{\frac{1}{b}}}{a} \]

11. Applied rewrites 99.5%

    \[\leadsto \left(\frac{0.5}{a + b} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\frac{\frac{1}{b}}{a}} \]

12. Final simplification 99.5%

    \[\leadsto \left(\frac{0.5}{a + b} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{{b}^{-1}}{a} \]
13. Add Preprocessing

Alternative 2: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\\ t_1 := -\mathsf{PI}\left(\right)\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{-0.5}{b}}{b \cdot a}\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b \cdot a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{t\_1}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* (- b a) (PI)) (* (* a b) (* (* 2.0 (+ a b)) (- b a)))))
        (t_1 (- (PI))))
   (if (<= b -8.5e+76)
     (/ (* t_1 (/ -0.5 b)) (* b a))
     (if (<= b -5.4e-107)
       t_0
       (if (<= b 5.6e-110)
         (* (/ (PI) a) (/ 0.5 (* b a)))
         (if (<= b 8.4e+46) t_0 (* (/ -0.5 b) (/ t_1 (* b a)))))))))

Derivation

1. Split input into 4 regimes

2. if b < -8.49999999999999992e76

   1. Initial program 70.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.5%

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

   12. Applied rewrites 98.6%

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

   if -8.49999999999999992e76 < b < -5.3999999999999999e-107 or 5.6000000000000001e-110 < b < 8.4e46

   1. Initial program 97.2%

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

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 95.1%

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

   if -5.3999999999999999e-107 < b < 5.6000000000000001e-110

   1. Initial program 74.5%

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

   3. 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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 91.8%

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

   if 8.4e46 < b

   1. Initial program 68.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.0%

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

Alternative 3: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{-0.5}{b} \cdot \frac{-\mathsf{PI}\left(\right)}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-77)
   (* (PI) (/ 0.5 (* (* a b) a)))
   (if (<= a 2.7e+34)
     (* (/ -0.5 b) (/ (- (PI)) (* b a)))
     (* (/ (PI) a) (/ 0.5 (* b a))))))

Derivation

1. Split input into 3 regimes

2. if a < -5.4000000000000001e-77

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 89.3%

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

   if -5.4000000000000001e-77 < a < 2.7e34

   1. Initial program 80.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 87.4%

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

   if 2.7e34 < a

   1. Initial program 74.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 95.4%

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

Alternative 4: 86.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-77)
   (* (PI) (/ 0.5 (* (* a b) a)))
   (if (<= a 2.7e+34)
     (/ (* (/ (PI) (* b a)) 0.5) b)
     (* (/ (PI) a) (/ 0.5 (* b a))))))

Derivation

1. Split input into 3 regimes

2. if a < -5.4000000000000001e-77

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 89.3%

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

   if -5.4000000000000001e-77 < a < 2.7e34

   1. Initial program 80.8%

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

   3. Taylor expanded in b around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 87.3%

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

   if 2.7e34 < a

   1. Initial program 74.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 95.4%

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

Alternative 5: 85.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot -0.5}{\left(b \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-77)
   (* (PI) (/ 0.5 (* (* a b) a)))
   (if (<= a 2.7e+34)
     (/ (* (- (PI)) -0.5) (* (* b a) b))
     (* (/ (PI) a) (/ 0.5 (* b a))))))

Derivation

1. Split input into 3 regimes

2. if a < -5.4000000000000001e-77

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 89.3%

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

   if -5.4000000000000001e-77 < a < 2.7e34

   1. Initial program 80.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 87.4%

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

   12. Applied rewrites 87.1%

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

   if 2.7e34 < a

   1. Initial program 74.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 95.4%

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

Alternative 6: 85.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot -0.5}{\left(b \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{\frac{0.5}{a}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-77)
   (* (PI) (/ 0.5 (* (* a b) a)))
   (if (<= a 2.7e+34)
     (/ (* (- (PI)) -0.5) (* (* b a) b))
     (* (PI) (/ (/ 0.5 a) (* a b))))))

Derivation

1. Split input into 3 regimes

2. if a < -5.4000000000000001e-77

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 89.3%

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

   if -5.4000000000000001e-77 < a < 2.7e34

   1. Initial program 80.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 87.4%

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

   12. Applied rewrites 87.1%

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

   if 2.7e34 < a

   1. Initial program 74.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 95.3%

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

Alternative 7: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(-\mathsf{PI}\left(\right)\right) \cdot -0.5}{\left(b \cdot a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-77)
   (* (PI) (/ 0.5 (* (* a b) a)))
   (if (<= a 2.3e+35)
     (/ (* (- (PI)) -0.5) (* (* b a) b))
     (* (/ (PI) (* (* b a) a)) 0.5))))

Derivation

1. Split input into 3 regimes

2. if a < -5.4000000000000001e-77

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 89.3%

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

   if -5.4000000000000001e-77 < a < 2.2999999999999998e35

   1. Initial program 81.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 67.1%

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

   7. Applied rewrites 80.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 86.8%

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

   12. Applied rewrites 86.5%

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

   if 2.2999999999999998e35 < a

   1. Initial program 73.7%

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

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 95.9%

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

Alternative 8: 80.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-77)
   (* (PI) (/ 0.5 (* (* a b) a)))
   (if (<= a 2.3e+35)
     (* (/ (PI) (* (* b b) a)) 0.5)
     (* (/ (PI) (* (* b a) a)) 0.5))))

Derivation

1. Split input into 3 regimes

2. if a < -5.4000000000000001e-77

   1. Initial program 81.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 80.3%

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

   9. Applied rewrites 89.3%

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

   if -5.4000000000000001e-77 < a < 2.2999999999999998e35

   1. Initial program 81.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. 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 73.3

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

   5. Applied rewrites 73.3%

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

   if 2.2999999999999998e35 < a

   1. Initial program 73.7%

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

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 95.9%

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

Alternative 9: 63.7% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 56.3

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

5. Applied rewrites 56.3%

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

7. Applied rewrites 62.1%

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

Alternative 10: 63.6% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 56.3

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

5. Applied rewrites 56.3%

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

7. Applied rewrites 56.3%

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

9. Applied rewrites 62.1%

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

Alternative 11: 57.7% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 56.3

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

5. Applied rewrites 56.3%

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

7. Applied rewrites 56.3%

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

Reproduce

herbie shell --seed 2024351 
(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))))