NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.3% → 99.6%

Time: 10.2s

Alternatives: 9

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: 78.3% 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, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.1%

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

4. Applied rewrites 90.4%

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-inverses N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. distribute-neg-frac N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-neg.f64 99.6

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.6

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

10. Applied rewrites 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(\left(a + b\right) \cdot a\right) \cdot \left(2 \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.1e+107)
   (* (/ 0.5 a) (/ (PI) (* a b)))
   (/ (PI) (* (* (+ a b) a) (* 2.0 b)))))

Derivation

1. Split input into 2 regimes

2. if a < -2.1e107

   1. Initial program 57.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 99.9%

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

   if -2.1e107 < a

   1. Initial program 86.2%

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

   4. Applied rewrites 92.6%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity 98.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(\left(a + b\right) \cdot a\right) \cdot \left(2 \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.8e+126)
   (* (/ (PI) a) (/ 0.5 (* a b)))
   (* (/ 0.5 (* (* (+ a b) a) b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -5.79999999999999971e126

   1. Initial program 53.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 77.9%

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

   9. Applied rewrites 99.8%

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

   if -5.79999999999999971e126 < a

   1. Initial program 86.4%

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

   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 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 99.6

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

   8. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 96.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 96.0

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

   10. Applied rewrites 96.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+138)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ 0.5 (* (* (+ a b) a) b)) (PI))))

Derivation

1. Split input into 2 regimes

2. if a < -2.0499999999999999e138

   1. Initial program 48.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.5

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 98.2%

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

   if -2.0499999999999999e138 < a

   1. Initial program 86.7%

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

   4. Applied rewrites 92.9%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 99.6

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

   8. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 96.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 96.0

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

   10. Applied rewrites 96.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\left(a + b\right) \cdot a\right) \cdot b} \cdot \mathsf{PI}\left(\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.1%

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

4. Applied rewrites 90.4%

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-inverses N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.1%

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

4. Applied rewrites 90.4%

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. *-inverses N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/r* N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.6

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

10. Applied rewrites 99.6%

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

11. Final simplification 99.6%

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

Alternative 7: 67.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.75e-107)
   (* (/ 0.5 (* (* a b) a)) (PI))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.74999999999999993e-107

   1. Initial program 78.8%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 79.0%

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

   if -1.74999999999999993e-107 < a

   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. 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 66.1

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

   5. Applied rewrites 66.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.5}{\left(a \cdot b\right) \cdot a} \cdot \mathsf{PI}\left(\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.8% 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 81.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 56.2

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

5. Applied rewrites 56.2%

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

7. Applied rewrites 56.2%

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

9. Applied rewrites 62.0%

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

10. Final simplification 62.0%

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

Alternative 9: 57.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.1%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 56.2

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

5. Applied rewrites 56.2%

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

7. Applied rewrites 56.2%

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

8. Final simplification 56.2%

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

Reproduce

herbie shell --seed 2024249 
(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))))