NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.7%

Time: 5.2s

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: 78.7% 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, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.0%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 88.2%

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

   1. lift-/.f64 N/A

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

   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

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 98.8

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 98.8

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/r* N/A

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

   9. lift-PI.f64 N/A

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

   10. lift--.f64 N/A

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

   11. associate-*r/ N/A

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

8. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+32} \lor \neg \left(a \leq 30000000000\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.5e+32) (not (<= a 30000000000.0)))
   (* (/ (PI) (* (* b a) a)) 0.5)
   (/ (* 0.5 (PI)) (* b (* b a)))))

Derivation

1. Split input into 2 regimes

2. if a < -5.49999999999999984e32 or 3e10 < a

   1. Initial program 75.7%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 91.6%

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

   if -5.49999999999999984e32 < a < 3e10

   1. Initial program 83.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \frac{a \cdot \mathsf{PI}\left(\right)}{{b}^{3}} + \frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{b}^{2}}}{a}} \]

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.8%

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

   10. Applied rewrites 87.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+32} \lor \neg \left(a \leq 30000000000\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+32} \lor \neg \left(a \leq 30000000000\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5.5e+32) (not (<= a 30000000000.0)))
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -5.49999999999999984e32 or 3e10 < a

   1. Initial program 75.7%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 91.6%

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

   if -5.49999999999999984e32 < a < 3e10

   1. Initial program 83.7%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 77.2%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+32} \lor \neg \left(a \leq 30000000000\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.0%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 88.2%

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

   1. lift-/.f64 N/A

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

   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

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 98.8

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 98.8

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. associate-/r* N/A

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

   9. lift-PI.f64 N/A

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

   10. lift--.f64 N/A

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

   11. associate-*r/ N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 98.9

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

10. Applied rewrites 98.9%

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

Alternative 5: 62.6% 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 80.0%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 51.2%

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

7. Applied rewrites 56.3%

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

Alternative 6: 56.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 51.2%

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

Alternative 7: 56.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.0%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 51.2%

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

7. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025018 
(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))))