NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 99.7%

Time: 6.0s

Alternatives: 6

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: 78.1% 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, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 79.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -550:\\ \;\;\;\;\frac{0.5}{\left(b \cdot a\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{elif}\;a \leq 430:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -550.0)
   (* (/ 0.5 (* (* b a) a)) (PI))
   (if (<= a 430.0)
     (* (/ (PI) (* (* b b) a)) 0.5)
     (/ (* (PI) 0.5) (* (* a b) a)))))

Derivation

1. Split input into 3 regimes

2. if a < -550

   1. Initial program 62.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.9%

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

   11. Applied rewrites 95.2%

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

   if -550 < a < 430

   1. Initial program 81.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   10. 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 78.7

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

   11. Applied rewrites 78.7%

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

   if 430 < a

   1. Initial program 81.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 80.5%

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

   9. Applied rewrites 90.8%

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

Alternative 3: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied rewrites 79.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/l/ N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 99.5

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.5

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 4: 63.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

7. Applied rewrites 55.9%

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

9. Applied rewrites 62.4%

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

Alternative 5: 63.0% 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 76.2%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

7. Applied rewrites 55.9%

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

9. Applied rewrites 55.9%

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

11. Applied rewrites 62.4%

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

Alternative 6: 57.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

7. Applied rewrites 55.9%

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

9. Applied rewrites 55.9%

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

Reproduce

herbie shell --seed 2025010 
(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))))