NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 9.0s

Alternatives: 17

Speedup: 1.8×

Specification

Math

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

FPCore

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

Initial Program: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. difference-of-squares N/A

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

   14. associate-*r* N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

4. Applied rewrites 85.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-PI.f64 N/A

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

   2. add-cbrt-cube N/A

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

   3. lower-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. add-cbrt-cube N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-pow.f64 99.4

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

8. Applied rewrites 99.4%

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

10. Applied rewrites 99.6%

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

Alternative 2: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(a + b\right)\\ t_1 := \frac{\frac{-1}{b}}{t\_0} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot \left(t\_0 \cdot \left(b - a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 2.0 (+ a b))) (t_1 (* (/ (/ -1.0 b) t_0) (/ (PI) (- b a)))))
   (if (<= a -3.3e-11)
     t_1
     (if (<= a 1.28e-188)
       (/ (/ (PI) b) (* 2.0 (* a b)))
       (if (<= a 4.2e+59)
         (/ (* (- b a) (PI)) (* (* a b) (* t_0 (- b a))))
         t_1)))))

Derivation

1. Split input into 3 regimes

2. if a < -3.3000000000000002e-11 or 4.19999999999999968e59 < 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. 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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 98.0%

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

   if -3.3000000000000002e-11 < a < 1.28e-188

   1. Initial program 74.3%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 1.28e-188 < a < 4.19999999999999968e59

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-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 92.8%

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

Alternative 3: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{-0.5}{a}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(b - a\right) \cdot \mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot \left(\left(2 \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ -0.5 a) b) (/ (PI) (- b a)))))
   (if (<= a -3.3e-11)
     t_0
     (if (<= a 1.28e-188)
       (/ (/ (PI) b) (* 2.0 (* a b)))
       (if (<= a 4.2e+59)
         (/ (* (- b a) (PI)) (* (* a b) (* (* 2.0 (+ a b)) (- b a))))
         t_0)))))

Derivation

1. Split input into 3 regimes

2. if a < -3.3000000000000002e-11 or 4.19999999999999968e59 < 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. 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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if -3.3000000000000002e-11 < a < 1.28e-188

   1. Initial program 74.3%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 1.28e-188 < a < 4.19999999999999968e59

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-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 92.8%

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

Alternative 4: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{-0.5}{a}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot \left(b - a\right)}{\left(b - a\right) \cdot \left(a \cdot \left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (/ (/ -0.5 a) b) (/ (PI) (- b a)))))
   (if (<= a -3.3e-11)
     t_0
     (if (<= a 1.28e-188)
       (/ (/ (PI) b) (* 2.0 (* a b)))
       (if (<= a 2e+61)
         (/ (* (PI) (- b a)) (* (- b a) (* a (* (* (+ a b) 2.0) b))))
         t_0)))))

Derivation

1. Split input into 3 regimes

2. if a < -3.3000000000000002e-11 or 1.9999999999999999e61 < 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. 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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if -3.3000000000000002e-11 < a < 1.28e-188

   1. Initial program 74.3%

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

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 1.28e-188 < a < 1.9999999999999999e61

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 92.7

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

   8. Applied rewrites 92.7%

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

Alternative 5: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. difference-of-squares N/A

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

   14. associate-*r* N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

4. Applied rewrites 85.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 6: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-11} \lor \neg \left(a \leq 8.5 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\frac{-0.5}{a}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.3e-11) (not (<= a 8.5e-39)))
   (* (/ (/ -0.5 a) b) (/ (PI) (- b a)))
   (/ (/ (PI) b) (* 2.0 (* a b)))))

Derivation

1. Split input into 2 regimes

2. if a < -3.3000000000000002e-11 or 8.5000000000000005e-39 < a

   1. Initial program 77.6%

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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -3.3000000000000002e-11 < a < 8.5000000000000005e-39

   1. Initial program 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-11} \lor \neg \left(a \leq 8.5 \cdot 10^{-39}\right):\\ \;\;\;\;\frac{\frac{-0.5}{a}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. difference-of-squares N/A

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

   14. associate-*r* N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

4. Applied rewrites 85.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

6. Applied rewrites 98.6%

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

Alternative 8: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

   1. lift-*.f64 N/A

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

   2. *-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

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. difference-of-squares N/A

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

   14. associate-*r* N/A

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

   15. *-lft-identity N/A

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

   16. *-rgt-identity N/A

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

4. Applied rewrites 85.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 98.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 98.6

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

6. Applied rewrites 98.6%

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

Alternative 9: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-18} \lor \neg \left(a \leq 1.55 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.8e-18) (not (<= a 1.55e-40)))
   (/ (- (PI)) (* (- b a) (* (* (+ a b) 2.0) b)))
   (/ (/ (PI) b) (* 2.0 (* a b)))))

Derivation

1. Split input into 2 regimes

2. if a < -5.8e-18 or 1.55000000000000005e-40 < a

   1. Initial program 77.6%

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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 97.4

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

   6. Applied rewrites 97.4%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-PI.f64 93.7

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

   9. Applied rewrites 93.7%

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

   if -5.8e-18 < a < 1.55000000000000005e-40

   1. Initial program 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 92.5

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

   7. Applied rewrites 92.5%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-18} \lor \neg \left(a \leq 1.55 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-46} \lor \neg \left(b \leq 8 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.65e-46) (not (<= b 8e-154)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))
   (/ (* (/ (PI) a) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if b < -1.65000000000000007e-46 or 7.9999999999999998e-154 < b

   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. 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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in a around 0

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

      1. lower-PI.f64 90.6

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

   9. Applied rewrites 90.6%

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

   if -1.65000000000000007e-46 < b < 7.9999999999999998e-154

   1. Initial program 70.4%

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

   3. Taylor expanded in a around 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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-46} \lor \neg \left(b \leq 8 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-46} \lor \neg \left(b \leq 8 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.65e-46) (not (<= b 8e-154)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))
   (* (/ 0.5 a) (/ (PI) (* a b)))))

Derivation

1. Split input into 2 regimes

2. if b < -1.65000000000000007e-46 or 7.9999999999999998e-154 < b

   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. 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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in a around 0

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

      1. lower-PI.f64 90.6

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

   9. Applied rewrites 90.6%

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

   if -1.65000000000000007e-46 < b < 7.9999999999999998e-154

   1. Initial program 70.4%

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

   3. Taylor expanded in a around 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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.2%

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

   9. Applied rewrites 91.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-46} \lor \neg \left(b \leq 8 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-8)
   (* (/ 0.5 a) (/ (PI) (* a b)))
   (if (<= a 6.6e+16)
     (/ (/ (PI) b) (* 2.0 (* a b)))
     (/ (* (/ (PI) a) 0.5) (* a b)))))

Derivation

1. Split input into 3 regimes

2. if a < -2e-8

   1. Initial program 73.0%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 78.9%

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

   9. Applied rewrites 94.5%

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

   if -2e-8 < a < 6.6e16

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if 6.6e16 < a

   1. Initial program 78.7%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 83.6

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

   5. Applied rewrites 83.6%

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

   7. Applied rewrites 94.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-46} \lor \neg \left(b \leq 8 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.65e-46) (not (<= b 8e-154)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))
   (/ (* (PI) 0.5) (* (* a b) a))))

Derivation

1. Split input into 2 regimes

2. if b < -1.65000000000000007e-46 or 7.9999999999999998e-154 < b

   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. 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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in a around 0

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

      1. lower-PI.f64 90.6

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

   9. Applied rewrites 90.6%

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

   if -1.65000000000000007e-46 < b < 7.9999999999999998e-154

   1. Initial program 70.4%

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

   3. Taylor expanded in a around 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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.2%

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

   9. Applied rewrites 90.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-46} \lor \neg \left(b \leq 8 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-8} \lor \neg \left(a \leq 1.5 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2e-8) (not (<= a 1.5e-38)))
   (/ (* (PI) 0.5) (* (* a b) a))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -2e-8 or 1.49999999999999994e-38 < a

   1. Initial program 77.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 89.6%

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

   if -2e-8 < a < 1.49999999999999994e-38

   1. Initial program 78.4%

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

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*r/ N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

      15. *-lft-identity N/A

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

      16. *-rgt-identity N/A

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

   4. Applied rewrites 68.6%

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

   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 75.9

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

   7. Applied rewrites 75.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-8} \lor \neg \left(a \leq 1.5 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.1% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   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 52.9

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

5. Applied rewrites 52.9%

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

7. Applied rewrites 53.0%

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

9. Applied rewrites 59.5%

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

10. Final simplification 59.5%

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

Alternative 16: 64.1% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   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 52.9

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

5. Applied rewrites 52.9%

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

7. Applied rewrites 53.0%

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

9. Applied rewrites 53.0%

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

11. Applied rewrites 59.4%

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

12. Final simplification 59.4%

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

Alternative 17: 58.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.9%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   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 52.9

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

5. Applied rewrites 52.9%

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

7. Applied rewrites 53.0%

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

9. Applied rewrites 53.0%

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

10. Final simplification 53.0%

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

Reproduce

herbie shell --seed 2024326 
(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))))