NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.7% → 99.7%

Time: 7.5s

Alternatives: 7

Speedup: 2.4×

Specification

Math

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

FPCore

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 89.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 98.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 98.4

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

8. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 89.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 98.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 98.4

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

8. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. associate-*l* N/A

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

   11. inv-pow N/A

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

   12. pow-plus N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2} \cdot {\left(b - a\right)}^{\color{blue}{0}}}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b} \]

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 69.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.4e-46)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6.3999999999999998e-46

   1. Initial program 71.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 93.3%

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

   if -6.3999999999999998e-46 < a

   1. Initial program 85.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   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 71.7

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

   5. 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. Add Preprocessing

Alternative 4: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 89.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift--.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 98.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 98.4

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

8. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. inv-pow N/A

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

   9. pow-plus N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot {\left(b - a\right)}^{\color{blue}{0}}\right)}{\left(a + b\right) \cdot \left(a \cdot b\right)} \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

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

Alternative 5: 62.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 62.8

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

5. Applied rewrites 62.8%

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

7. Applied rewrites 69.3%

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

Alternative 6: 62.8% 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 80.9%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 62.8

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

5. Applied rewrites 62.8%

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

7. Applied rewrites 62.8%

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

9. Applied rewrites 69.4%

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

10. Final simplification 69.4%

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

Alternative 7: 56.9% 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 80.9%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 62.8

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

5. Applied rewrites 62.8%

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

7. Applied rewrites 62.8%

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

8. Final simplification 62.8%

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

Reproduce

herbie shell --seed 2024331 
(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))))