NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.4% → 99.7%

Time: 8.2s

Alternatives: 8

Speedup: 2.4×

Specification

Math

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

FPCore

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 90.5%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 90.5%

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot 1\right)}{a + b}} \]
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. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 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%

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

Alternative 3: 72.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.4e-13)
   (* (/ (PI) (* (* b a) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if b < 8.39999999999999955e-13

   1. Initial program 80.8%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 65.4

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

   5. Applied rewrites 65.4%

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

   7. Applied rewrites 72.8%

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

   if 8.39999999999999955e-13 < b

   1. Initial program 74.9%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot a\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 90.5%

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot 1\right)}{a + b}} \]
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. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.2

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.2

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 99.2

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

10. Applied rewrites 99.2%

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

11. Final simplification 99.2%

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

Alternative 5: 93.3% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.3%

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

4. Applied rewrites 90.5%

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot 1\right)}{a + b}} \]
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. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 92.6

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 92.6

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

10. Applied rewrites 92.6%

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

11. Final simplification 92.6%

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

Alternative 6: 63.5% 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 79.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.2

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

5. Applied rewrites 58.2%

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

7. Applied rewrites 63.7%

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

8. Final simplification 63.7%

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

Alternative 7: 57.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.2

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

5. Applied rewrites 58.2%

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

6. Final simplification 58.2%

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

Alternative 8: 57.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.2

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

5. Applied rewrites 58.2%

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

7. Applied rewrites 58.2%

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

8. Final simplification 58.2%

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

Reproduce

herbie shell --seed 2024255 
(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))))