NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.7%

Time: 5.0s

Alternatives: 5

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: 78.5% 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

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

FPCore

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

Derivation

1. Initial program 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 88.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.6

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.7%

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

Alternative 2: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+23} \lor \neg \left(b \leq 1.3 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= b -1.22e+23) (not (<= b 1.3e+36)))
   (* (/ (PI) (* (* b b) a)) 0.5)
   (* (PI) (/ 0.5 (* (* a b) a)))))

Derivation

1. Split input into 2 regimes

2. if b < -1.22e23 or 1.3000000000000001e36 < b

   1. Initial program 78.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

   if -1.22e23 < b < 1.3000000000000001e36

   1. Initial program 85.3%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
   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. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 80.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+23} \lor \neg \left(b \leq 1.3 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(a \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-sub N/A

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

   9. frac-times N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 88.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.6

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 98.6

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 98.6

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

10. Applied rewrites 98.6%

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

Alternative 4: 62.7% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
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. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

7. Applied rewrites 54.3%

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

9. Applied rewrites 59.6%

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

Alternative 5: 57.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{{a}^{2} \cdot b}} \]
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. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 54.2

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

5. Applied rewrites 54.2%

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

7. Applied rewrites 54.3%

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

Reproduce

herbie shell --seed 2025017 
(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))))