NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.9% → 99.6%

Time: 4.9s

Alternatives: 8

Speedup: 1.8×

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: 78.9% 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

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

FPCore

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

Derivation

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. 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

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. difference-of-squares N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 88.6

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

4. Applied rewrites 88.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow-1 N/A

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

    4. *-commutative N/A

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

    5. lower-/.f64 N/A

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

    6. *-commutative N/A

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

    7. lift-*.f64 99.7

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

11. Applied rewrites 99.7%

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

Alternative 2: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot b\\ t_1 := \frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot b\right) \cdot \left(b + a\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-220}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{t\_0}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (PI) b)) (t_1 (* (/ (PI) (* b (* b a))) 0.5)))
   (if (<= b -3.1e+139)
     t_1
     (if (<= b -6.4e-157)
       (/ t_0 (* (* (* (* a 2.0) b) b) (+ b a)))
       (if (<= b 4e-220)
         (* (/ (PI) (* a (* a b))) 0.5)
         (if (<= b 2.2e+75) (/ t_0 (* (* a b) (* 2.0 (* (+ b a) b)))) t_1))))))

Derivation

1. Split input into 4 regimes

2. if b < -3.1e139 or 2.20000000000000012e75 < b

   1. Initial program 58.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   8. Taylor expanded in a around 0

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. difference-of-squares-rev N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

      6. frac-sub N/A

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

      7. *-lft-identity N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 81.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 98.9

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

   12. Applied rewrites 98.9%

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

   if -3.1e139 < b < -6.40000000000000041e-157

   1. Initial program 94.0%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 69.2

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

   7. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift--.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 69.2

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

   9. Applied rewrites 69.2%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 95.2%

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

   if -6.40000000000000041e-157 < b < 3.99999999999999997e-220

   1. Initial program 64.9%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 3.99999999999999997e-220 < b < 2.20000000000000012e75

   1. Initial program 96.4%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 62.9

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 91.5%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot b}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot b\right) \cdot \left(b + a\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-220}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ t_1 := \frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{if}\;b \leq -2.55 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-220}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* (PI) b) (* (* a b) (* 2.0 (* (+ b a) b)))))
        (t_1 (* (/ (PI) (* b (* b a))) 0.5)))
   (if (<= b -2.55e+73)
     t_1
     (if (<= b -6.4e-157)
       t_0
       (if (<= b 4e-220)
         (* (/ (PI) (* a (* a b))) 0.5)
         (if (<= b 2.2e+75) t_0 t_1))))))

Derivation

1. Split input into 3 regimes

2. if b < -2.55000000000000012e73 or 2.20000000000000012e75 < b

   1. Initial program 64.1%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.2

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

   5. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in a around 0

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. difference-of-squares-rev N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

      6. frac-sub N/A

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

      7. *-lft-identity N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 83.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 99.0

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

   12. Applied rewrites 99.0%

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

   if -2.55000000000000012e73 < b < -6.40000000000000041e-157 or 3.99999999999999997e-220 < b < 2.20000000000000012e75

   1. Initial program 94.4%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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. 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-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-PI.f64 63.1

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.1%

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

   if -6.40000000000000041e-157 < b < 3.99999999999999997e-220

   1. Initial program 64.9%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+73}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-157}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-220}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20} \lor \neg \left(a \leq 8.8 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6e-20) (not (<= a 8.8e-73)))
   (* (/ (/ (PI) a) (* b a)) 0.5)
   (* (/ (PI) (* b (* b a))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000057e-20 or 8.8000000000000001e-73 < a

   1. Initial program 77.6%

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

   3. Taylor expanded in a around 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 89.9

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

   9. Applied rewrites 89.9%

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

   if -6.00000000000000057e-20 < a < 8.8000000000000001e-73

   1. Initial program 76.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 32.6

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in a around 0

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. difference-of-squares-rev N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

      6. frac-sub N/A

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

      7. *-lft-identity N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 75.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 88.9

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

   12. Applied rewrites 88.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20} \lor \neg \left(a \leq 8.8 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a}}{a}\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6e-20)
   (/ (/ (* (/ (PI) b) 0.5) a) a)
   (if (<= a 8.8e-73)
     (* (/ (PI) (* b (* b a))) 0.5)
     (* (/ (/ (PI) a) (* b a)) 0.5))))

Derivation

1. Split input into 3 regimes

2. if a < -6.00000000000000057e-20

   1. Initial program 81.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-/.f64 N/A

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 92.7

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

   10. Applied rewrites 92.7%

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

   if -6.00000000000000057e-20 < a < 8.8000000000000001e-73

   1. Initial program 76.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 32.6

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in a around 0

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. difference-of-squares-rev N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

      6. frac-sub N/A

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

      7. *-lft-identity N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 75.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 88.9

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

   12. Applied rewrites 88.9%

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

   if 8.8000000000000001e-73 < a

   1. Initial program 74.2%

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

   3. Taylor expanded in 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 86.5

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

   7. Applied rewrites 86.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 87.8

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

   9. Applied rewrites 87.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{b} \cdot 0.5}{a}}{a}\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20} \lor \neg \left(a \leq 8.8 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6e-20) (not (<= a 8.8e-73)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* b (* b a))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000057e-20 or 8.8000000000000001e-73 < a

   1. Initial program 77.6%

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

   3. Taylor expanded in a around 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

   if -6.00000000000000057e-20 < a < 8.8000000000000001e-73

   1. Initial program 76.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 32.5

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

   5. Applied rewrites 32.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 32.6

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in a around 0

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

      1. frac-times N/A

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

      2. *-rgt-identity N/A

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

      3. difference-of-squares-rev N/A

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

      6. frac-sub N/A

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

      7. *-lft-identity N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   10. Applied rewrites 75.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 88.9

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

   12. Applied rewrites 88.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20} \lor \neg \left(a \leq 8.8 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20} \lor \neg \left(a \leq 8.8 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -6e-20) (not (<= a 8.8e-73)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000057e-20 or 8.8000000000000001e-73 < a

   1. Initial program 77.6%

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

   3. Taylor expanded in a around 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

   if -6.00000000000000057e-20 < a < 8.8000000000000001e-73

   1. Initial program 76.7%

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

   3. Taylor expanded in a around 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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-20} \lor \neg \left(a \leq 8.8 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

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 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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 60.6

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

5. Applied rewrites 60.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 66.0

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

7. Applied rewrites 66.0%

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

8. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2025037 
(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))))