NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.0% → 99.6%

Time: 7.9s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

Initial Program: 79.0% 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

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

FPCore

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

Derivation

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. *-commutative N/A

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

   4. *-rgt-identity N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

4. Applied rewrites 84.8%

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

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift--.f64 N/A

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

   14. difference-of-squares N/A

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

   15. times-frac N/A

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

   16. difference-of-squares N/A

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

6. Applied rewrites 99.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.6

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.6

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 2: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(a \cdot b\right)\\ t_1 := \frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{t\_0}\\ \mathbf{if}\;a \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{elif}\;a \leq -1.85 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{t\_0}\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 2.0 (* a b)))
        (t_1 (/ (* (/ (PI) (* (+ a b) (- b a))) (- b a)) t_0)))
   (if (<= a -2e+94)
     (/ (* (/ 0.5 a) (PI)) (* a b))
     (if (<= a -1.85e-67)
       t_1
       (if (<= a 3.2e-125)
         (/ (/ (PI) b) t_0)
         (if (<= a 1.32e+45) t_1 (* (/ (/ 0.5 a) (* a b)) (PI))))))))

Derivation

1. Split input into 4 regimes

2. if a < -2e94

   1. Initial program 62.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.4%

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

   9. Applied rewrites 98.6%

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

   11. Applied rewrites 99.8%

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

   if -2e94 < a < -1.85e-67 or 3.1999999999999998e-125 < a < 1.32000000000000005e45

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 95.3%

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

   if -1.85e-67 < a < 3.1999999999999998e-125

   1. Initial program 71.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 79.1%

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

   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 96.3

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

   7. Applied rewrites 96.3%

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

   if 1.32000000000000005e45 < a

   1. Initial program 65.4%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 85.0

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

   5. Applied rewrites 85.0%

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 99.8%

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

   11. Applied rewrites 99.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{elif}\;a \leq -1.85 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 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. *-commutative N/A

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

   4. *-rgt-identity N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

4. Applied rewrites 84.8%

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

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift--.f64 N/A

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

   14. difference-of-squares N/A

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

   15. times-frac N/A

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

   16. difference-of-squares N/A

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

6. Applied rewrites 99.0%

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

Alternative 4: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot a\right)}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.28e-70) (not (<= b 2.55e-23)))
   (/ (/ (PI) (* 2.0 (* b a))) (- b a))
   (* (/ (/ 0.5 a) (* a b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if b < -1.28e-70 or 2.55000000000000005e-23 < b

   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. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 74.7%

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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. times-frac N/A

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

      16. difference-of-squares N/A

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

   6. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-PI.f64 94.7

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

   11. Applied rewrites 94.7%

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

   if -1.28e-70 < b < 2.55000000000000005e-23

   1. Initial program 76.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 90.2%

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

   11. Applied rewrites 90.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{2 \cdot \left(b \cdot a\right)}}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.28e-70) (not (<= b 2.55e-23)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))
   (* (/ (/ 0.5 a) (* a b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if b < -1.28e-70 or 2.55000000000000005e-23 < b

   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. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 74.7%

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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. times-frac N/A

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

      16. difference-of-squares N/A

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

   6. Applied rewrites 98.9%

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

   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 94.0

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

   9. Applied rewrites 94.0%

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

   if -1.28e-70 < b < 2.55000000000000005e-23

   1. Initial program 76.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 90.2%

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

   11. Applied rewrites 90.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a}}{a \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot \frac{0.5}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.28e-70) (not (<= b 2.55e-23)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))
   (* (/ (PI) (* b a)) (/ 0.5 a))))

Derivation

1. Split input into 2 regimes

2. if b < -1.28e-70 or 2.55000000000000005e-23 < b

   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. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 74.7%

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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. times-frac N/A

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

      16. difference-of-squares N/A

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

   6. Applied rewrites 98.9%

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

   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 94.0

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

   9. Applied rewrites 94.0%

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

   if -1.28e-70 < b < 2.55000000000000005e-23

   1. Initial program 76.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 90.2%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot a} \cdot \frac{0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(b \cdot a\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.28e-70) (not (<= b 2.55e-23)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))
   (/ (* (PI) 0.5) (* (* b a) a))))

Derivation

1. Split input into 2 regimes

2. if b < -1.28e-70 or 2.55000000000000005e-23 < b

   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. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 74.7%

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

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift--.f64 N/A

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

      14. difference-of-squares N/A

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

      15. times-frac N/A

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

      16. difference-of-squares N/A

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

   6. Applied rewrites 98.9%

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

   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 94.0

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

   9. Applied rewrites 94.0%

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

   if -1.28e-70 < b < 2.55000000000000005e-23

   1. Initial program 76.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 73.7%

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

   9. Applied rewrites 89.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{-70} \lor \neg \left(b \leq 2.55 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(b \cdot a\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+16} \lor \neg \left(a \leq 2.6 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(b \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.2e+16) (not (<= a 2.6e+28)))
   (/ (* (PI) 0.5) (* (* b a) a))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -9.2e16 or 2.6000000000000002e28 < a

   1. Initial program 69.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 95.6%

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

   if -9.2e16 < a < 2.6000000000000002e28

   1. Initial program 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in a around 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 71.7

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

   7. Applied rewrites 71.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+16} \lor \neg \left(a \leq 2.6 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(b \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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. 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 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 56.1%

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

9. Applied rewrites 62.6%

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

10. Final simplification 62.6%

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

Alternative 10: 56.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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. 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 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 56.1%

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

8. Final simplification 56.1%

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

Alternative 11: 56.3% 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 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. 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 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 56.1%

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

9. Applied rewrites 62.6%

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

11. Applied rewrites 56.0%

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

12. Final simplification 56.0%

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

Reproduce

herbie shell --seed 2024358 
(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))))