NMSE Section 6.1 mentioned, B

Percentage Accurate: 79.0% → 99.7%

Time: 7.4s

Alternatives: 7

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: 79.0% 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

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

FPCore

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

Derivation

1. Initial program 77.3%

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

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 89.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.3%

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

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 89.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 68.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4e+31)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -3.9999999999999999e31

   1. Initial program 64.7%

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

   3. Taylor expanded in 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 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 94.7%

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

   if -3.9999999999999999e31 < a

   1. Initial program 80.6%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.3%

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

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 89.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. lift-PI.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 98.2

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.2

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

10. Applied rewrites 98.2%

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

11. Final simplification 98.2%

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

Alternative 5: 62.6% 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 77.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.1

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

5. Applied rewrites 58.1%

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

7. Applied rewrites 62.8%

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

8. Final simplification 62.8%

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

Alternative 6: 62.6% 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 77.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.1

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

5. Applied rewrites 58.1%

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

7. Applied rewrites 58.1%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{0.5}{\left(a \cdot a\right) \cdot b}} \]
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 7: 56.8% 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 77.3%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 58.1

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

5. Applied rewrites 58.1%

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

7. Applied rewrites 58.1%

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

8. Final simplification 58.1%

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

Reproduce

herbie shell --seed 2024304 
(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))))