NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.7% → 99.6%

Time: 9.1s

Alternatives: 12

Speedup: 2.0×

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: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.2%

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

4. Applied rewrites 89.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

6. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

   9. inv-pow N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-rgt-identity N/A

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

   14. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.5\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{t\_0}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (PI) 0.5)))
   (if (<= a -1.95e+96) (/ (/ t_0 (* b a)) a) (/ t_0 (* (* (+ b a) a) b)))))

Derivation

1. Split input into 2 regimes

2. if a < -1.95e96

   1. Initial program 60.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 99.7%

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

   if -1.95e96 < a

   1. Initial program 81.9%

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

   4. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. inv-pow N/A

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

      12. pow-plus N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 95.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right) \cdot 0.5}{b \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e+93)
   (* (/ 0.5 a) (/ (PI) (* b a)))
   (/ (* (PI) 0.5) (* (* (+ b a) a) b))))

Derivation

1. Split input into 2 regimes

2. if a < -3.10000000000000019e93

   1. Initial program 60.8%

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

   3. Taylor expanded in 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 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 99.8%

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

   if -3.10000000000000019e93 < a

   1. Initial program 81.8%

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

   4. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. inv-pow N/A

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

      12. pow-plus N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 95.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+96)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (/ (* (PI) 0.5) (* (* (+ b a) a) b))))

Derivation

1. Split input into 2 regimes

2. if a < -2.0000000000000001e96

   1. Initial program 60.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 95.5%

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

   if -2.0000000000000001e96 < a

   1. Initial program 81.9%

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

   4. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. inv-pow N/A

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

      12. pow-plus N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 95.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+96)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ 0.5 (* (* (+ b a) a) b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -3.29999999999999984e96

   1. Initial program 60.1%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 95.5%

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

   if -3.29999999999999984e96 < a

   1. Initial program 81.9%

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

   4. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. inv-pow N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. *-rgt-identity 99.6

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

   8. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lift-+.f64 N/A

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

      13. lower-*.f64 95.5

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 95.5

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

   10. Applied rewrites 95.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(b + a\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.2%

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

4. Applied rewrites 89.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{a + b}, \frac{0.5}{b - a} \cdot {a}^{-1}, \frac{\frac{0.5}{a + b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}}{-b}\right)} \]

6. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. inv-pow N/A

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

   6. pow-plus N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. *-rgt-identity 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 7: 74.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-32)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ (PI) (* (* b a) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e-32

   1. Initial program 71.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 90.4%

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

   if -2.3000000000000001e-32 < a

   1. Initial program 79.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      4. un-div-inv N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. frac-2neg N/A

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

      11. distribute-lft-neg-out N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-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 59.3

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 67.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot b\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-32)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ 0.5 (* (* b b) a)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e-32

   1. Initial program 71.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 90.4%

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

   if -2.3000000000000001e-32 < a

   1. Initial program 79.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

      4. un-div-inv N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. frac-2neg N/A

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

      11. distribute-lft-neg-out N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-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 59.3

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 59.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(b \cdot b\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-32)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e-32

   1. Initial program 71.3%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 90.4%

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

   if -2.3000000000000001e-32 < a

   1. Initial program 79.7%

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

   3. Taylor expanded in 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 59.3

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

   5. Applied rewrites 59.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% 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 77.2%

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

3. Taylor expanded in 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 58.3

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

5. Applied rewrites 58.3%

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

7. Applied rewrites 65.6%

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

8. Final simplification 65.6%

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

Alternative 11: 63.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in 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 58.3

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

5. Applied rewrites 58.3%

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

7. Applied rewrites 58.3%

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

9. Applied rewrites 65.5%

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

10. Final simplification 65.5%

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

Alternative 12: 57.2% 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 77.2%

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

3. Taylor expanded in 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 58.3

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

5. Applied rewrites 58.3%

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

7. Applied rewrites 58.3%

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

8. Final simplification 58.3%

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

Reproduce

herbie shell --seed 2024248 
(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))))