NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.0% → 99.6%

Time: 8.8s

Alternatives: 8

Speedup: 2.4×

Specification

Math

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

FPCore

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

Initial Program: 78.0% 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, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.0%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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. 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 90.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift--.f64 N/A

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

   8. difference-of-squares N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. difference-of-squares N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 90.3

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 90.3

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

6. Applied rewrites 90.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. *-inverses N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.0%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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. 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 90.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift--.f64 N/A

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

   8. difference-of-squares N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. difference-of-squares N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 90.3

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 90.3

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

6. Applied rewrites 90.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. *-inverses N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.7

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. *-inverses N/A

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

   9. clear-num N/A

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

   10. times-frac N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-PI.f64 N/A

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

   14. associate-*l/ N/A

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

10. Applied rewrites 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 75.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.105:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.105)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (* (/ (PI) (* (* b a) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -0.104999999999999996

   1. Initial program 87.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   7. Applied rewrites 87.3%

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

   9. Applied rewrites 95.1%

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

   if -0.104999999999999996 < a

   1. Initial program 80.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 90.2%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   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. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.105:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.12e-39)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (* (/ 0.5 (* (* b a) b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -1.12e-39

   1. Initial program 88.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 91.0%

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

   if -1.12e-39 < a

   1. Initial program 80.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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 89.9%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   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. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.4

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

   7. Applied rewrites 71.4%

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

   9. Applied rewrites 71.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.12e-39)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.12e-39

   1. Initial program 88.0%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 91.0%

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

   if -1.12e-39 < a

   1. Initial program 80.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{-39}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.0%

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

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 90.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 7: 63.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 61.4

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

5. Applied rewrites 61.4%

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

7. Applied rewrites 61.4%

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

9. Applied rewrites 66.1%

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

10. Final simplification 66.1%

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

Alternative 8: 57.5% 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 82.0%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 61.4

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

5. Applied rewrites 61.4%

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

7. Applied rewrites 61.4%

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

8. Final simplification 61.4%

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

Reproduce

herbie shell --seed 2024254 
(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))))