NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.6%

Time: 9.2s

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: 78.2% 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, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{PI}\left(\right) \cdot 0.5}{a + b} \cdot \frac{1}{a \cdot b} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (/ (* (PI) 0.5) (+ a b)) (/ 1.0 (* a b))))

Derivation

1. Initial program 74.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. *-lft-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 86.7%

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

5. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right) \cdot 0.5}{a + b} \cdot \frac{1}{a \cdot b} \]
9. Add Preprocessing

Alternative 2: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot b}}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-105)
   (/ (/ (* (PI) 0.5) a) (* a b))
   (* (/ (/ (PI) (* a b)) b) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999993e-105

   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 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 78.7%

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

   if -1.99999999999999993e-105 < a

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. difference-of-squares N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l* N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   8. 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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.0

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

   9. Applied rewrites 74.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a \cdot b}}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right) \cdot 0.5}{a}}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-105)
   (/ (/ (* (PI) 0.5) a) (* a b))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999993e-105

   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 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 78.7%

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

   if -1.99999999999999993e-105 < a

   1. Initial program 73.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

Alternative 4: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-105)
   (* (/ (PI) a) (/ 0.5 (* a b)))
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999993e-105

   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 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 78.7%

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

   if -1.99999999999999993e-105 < a

   1. Initial program 73.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. un-div-inv N/A

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

   6. associate-*r/ N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. difference-of-squares N/A

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

   11. *-commutative N/A

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

   12. *-rgt-identity N/A

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

   13. *-lft-identity N/A

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

   14. times-frac N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 86.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. frac-times N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*l* N/A

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot 1\right)}{a + b}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-rgt-identity N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.6

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 6: 67.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-105)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.99999999999999993e-105

   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 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 79.1%

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

   if -1.99999999999999993e-105 < a

   1. Initial program 73.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

Alternative 7: 63.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.6

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

5. Applied rewrites 57.6%

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

7. Applied rewrites 62.8%

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

Alternative 8: 63.3% 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 74.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.6

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

5. Applied rewrites 57.6%

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

7. Applied rewrites 57.6%

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

9. Applied rewrites 62.8%

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

10. Final simplification 62.8%

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

Alternative 9: 57.4% 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 74.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.6

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

5. Applied rewrites 57.6%

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

7. Applied rewrites 57.6%

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

8. Final simplification 57.6%

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

Reproduce

herbie shell --seed 2024270 
(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))))