NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.6%

Time: 8.8s

Alternatives: 12

Speedup: 1.4×

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: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

   1. lift-*.f64 N/A

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

   2. *-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 88.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. frac-times N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 77.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot \left(b - a\right)}{\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-136)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (if (<= b 3.3e+83)
     (/ (* (PI) (- b a)) (* (* (+ a b) (- b a)) (* 2.0 (* a b))))
     (/ (* (/ (PI) b) 0.5) (* a b)))))

Derivation

1. Split input into 3 regimes

2. if b < 3.50000000000000029e-136

   1. Initial program 77.1%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 68.9%

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

   if 3.50000000000000029e-136 < b < 3.29999999999999985e83

   1. Initial program 94.9%

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

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 88.6%

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

   if 3.29999999999999985e83 < b

   1. Initial program 72.0%

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

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot \left(b - a\right)}{\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(2 \cdot \left(a \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

   1. lift-*.f64 N/A

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

   2. *-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 88.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. frac-times N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

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

Alternative 4: 76.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{\frac{0.5}{b - a}}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-66)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (PI) (/ (/ 0.5 (- b a)) (* a b)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.29999999999999992e-66

   1. Initial program 78.4%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 69.9%

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

   if 2.29999999999999992e-66 < b

   1. Initial program 82.8%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in b around inf

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

       1. lower-PI.f64 92.7

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

   11. Applied rewrites 92.7%

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

Alternative 5: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e-62)
   (/ (* (/ (PI) a) 0.5) (* a b))
   (/ (* (/ (PI) b) 0.5) (* a b))))

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999986e-62

   1. Initial program 76.6%

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

      1. lift-*.f64 N/A

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

      2. *-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

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if -2.89999999999999986e-62 < a

   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

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 70.8

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

   7. Applied rewrites 70.8%

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

Alternative 6: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \mathsf{PI}\left(\right)}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e-62)
   (/ (* (/ (PI) a) 0.5) (* a b))
   (/ (* (/ 0.5 a) (PI)) (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999986e-62

   1. Initial program 76.6%

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

      1. lift-*.f64 N/A

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

      2. *-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

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-sub N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if -2.89999999999999986e-62 < a

   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 b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 62.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \mathsf{PI}\left(\right)}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \mathsf{PI}\left(\right)}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e-62)
   (* (/ 0.5 (* a b)) (/ (PI) a))
   (/ (* (/ 0.5 a) (PI)) (* b b))))

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999986e-62

   1. Initial program 76.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 74.9%

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

   9. Applied rewrites 89.2%

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

   if -2.89999999999999986e-62 < a

   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 b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 62.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \mathsf{PI}\left(\right)}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e-62)
   (* (/ 0.5 (* a b)) (/ (PI) a))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999986e-62

   1. Initial program 76.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 74.9%

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

   9. Applied rewrites 89.2%

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

   if -2.89999999999999986e-62 < a

   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 b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.3

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

   5. Applied rewrites 62.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{0.5}{a \cdot b} \cdot \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e-62)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -2.89999999999999986e-62

   1. Initial program 76.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 88.7%

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

   if -2.89999999999999986e-62 < a

   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 b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 10: 62.4% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

7. Applied rewrites 63.8%

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

Alternative 11: 62.4% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

7. Applied rewrites 56.6%

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

9. Applied rewrites 63.8%

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

10. Final simplification 63.8%

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

Alternative 12: 56.6% 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 79.5%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

7. Applied rewrites 56.6%

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

8. Final simplification 56.6%

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

Reproduce

herbie shell --seed 2024276 
(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))))