NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.1% → 98.7%

Time: 7.6s

Alternatives: 13

Speedup: 1.6×

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: 78.1% 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: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.5}{a}}{b}\\ \mathbf{if}\;b \leq -7.5 \cdot 10^{+85}:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{PI}\left(\right)}{b}\\ \mathbf{elif}\;b \leq 1.48 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{\frac{b - a}{a}}{b}}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{PI}\left(\right)}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ 0.5 a) b)))
   (if (<= b -7.5e+85)
     (* t_0 (/ (PI) b))
     (if (<= b 1.48e+59)
       (* (/ (/ (/ (- b a) a) b) (* 2.0 (+ a b))) (/ (PI) (- b a)))
       (/ (* t_0 (PI)) (- b a))))))

Derivation

1. Split input into 3 regimes

2. if b < -7.49999999999999942e85

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-PI.f64 99.8

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

   10. Applied rewrites 99.8%

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

   if -7.49999999999999942e85 < b < 1.4800000000000001e59

   1. Initial program 84.9%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 98.5%

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

   if 1.4800000000000001e59 < b

   1. Initial program 63.0%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 65.9%

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

   5. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 98.6

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

   9. Applied rewrites 98.6%

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

Alternative 2: 96.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{2 \cdot \left(a \cdot b\right)}\\ t_1 := \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{t\_1 \cdot 0.5}{b}}{b}\\ \mathbf{elif}\;b \leq -4.05 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{0.5}{b}}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* (/ (PI) (* (+ a b) (- b a))) (- b a)) (* 2.0 (* a b))))
        (t_1 (/ (PI) a)))
   (if (<= b -1.5e+88)
     (/ (/ (* t_1 0.5) b) b)
     (if (<= b -4.05e-109)
       t_0
       (if (<= b 1.35e-103)
         (/ (* t_1 (/ 0.5 b)) a)
         (if (<= b 5e+106) t_0 (/ (* (/ 0.5 b) (PI)) (* a b))))))))

Derivation

1. Split input into 4 regimes

2. if b < -1.50000000000000003e88

   1. Initial program 55.6%

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

      1. lift-*.f64 N/A

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

      2. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 75.1

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 99.8%

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

   if -1.50000000000000003e88 < b < -4.0500000000000001e-109 or 1.35000000000000005e-103 < b < 4.9999999999999998e106

   1. Initial program 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-sub N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 97.6%

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

   if -4.0500000000000001e-109 < b < 1.35000000000000005e-103

   1. Initial program 73.9%

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

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 95.2%

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

   if 4.9999999999999998e106 < b

   1. Initial program 56.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 69.0

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

   7. Applied rewrites 69.0%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b}}{b}\\ \mathbf{elif}\;b \leq -4.05 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{b}}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{\left(a + b\right) \cdot \left(b - a\right)} \cdot \left(b - a\right)}{2 \cdot \left(a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.5}{a}}{b}\\ \mathbf{if}\;b \leq -5.1 \cdot 10^{+69}:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{elif}\;b \leq 1.48 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{b - a}{a} \cdot \mathsf{PI}\left(\right)}{\left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right) \cdot \left(b - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{PI}\left(\right)}{b - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (/ 0.5 a) b)))
   (if (<= b -5.1e+69)
     (* t_0 (/ (PI) (- b a)))
     (if (<= b 1.48e+59)
       (/ (* (/ (- b a) a) (PI)) (* (* (* (+ a b) 2.0) b) (- b a)))
       (/ (* t_0 (PI)) (- b a))))))

Derivation

1. Split input into 3 regimes

2. if b < -5.09999999999999999e69

   1. Initial program 61.2%

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

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -5.09999999999999999e69 < b < 1.4800000000000001e59

   1. Initial program 84.5%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 97.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if 1.4800000000000001e59 < b

   1. Initial program 63.0%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 65.9%

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

   5. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.5

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

   7. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 98.6

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

   9. Applied rewrites 98.6%

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

Alternative 4: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{if}\;a \leq -0.00125:\\ \;\;\;\;\frac{\frac{-0.5}{a}}{b} \cdot t\_0\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{b}}{2 \cdot \left(a + b\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (PI) (- b a))))
   (if (<= a -0.00125)
     (* (/ (/ -0.5 a) b) t_0)
     (if (<= a 1.1e-132)
       (/ (/ (* (/ (PI) a) 0.5) b) b)
       (* (/ (/ -1.0 b) (* 2.0 (+ a b))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if a < -0.00125000000000000003

   1. Initial program 74.5%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if -0.00125000000000000003 < a < 1.09999999999999995e-132

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.4

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

   7. Applied rewrites 68.4%

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

   9. Applied rewrites 91.2%

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

   if 1.09999999999999995e-132 < a

   1. Initial program 85.5%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 92.2%

      \[\leadsto \frac{\frac{\color{blue}{-1}}{b}}{2 \cdot \left(a + b\right)} \cdot \frac{\mathsf{PI}\left(\right)}{b - a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00125 \lor \neg \left(a \leq 1.1 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{\frac{-0.5}{a}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.00125) (not (<= a 1.1e-132)))
   (* (/ (/ -0.5 a) b) (/ (PI) (- b a)))
   (/ (/ (* (/ (PI) a) 0.5) b) b)))

Derivation

1. Split input into 2 regimes

2. if a < -0.00125000000000000003 or 1.09999999999999995e-132 < a

   1. Initial program 80.9%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 93.8

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

   7. Applied rewrites 93.8%

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

   if -0.00125000000000000003 < a < 1.09999999999999995e-132

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.4

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

   7. Applied rewrites 68.4%

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

   9. Applied rewrites 91.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00125 \lor \neg \left(a \leq 1.1 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{\frac{-0.5}{a}}{b} \cdot \frac{\mathsf{PI}\left(\right)}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{\left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right) \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0245)
   (* (/ (PI) (* a b)) (/ 0.5 a))
   (if (<= a 1.1e-132)
     (/ (/ (* (/ (PI) a) 0.5) b) b)
     (/ (- (PI)) (* (* (* (+ a b) 2.0) b) (- b a))))))

Derivation

1. Split input into 3 regimes

2. if a < -0.024500000000000001

   1. Initial program 74.5%

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

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 78.8%

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

   9. Applied rewrites 95.2%

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

   if -0.024500000000000001 < a < 1.09999999999999995e-132

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.4

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

   7. Applied rewrites 68.4%

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

   9. Applied rewrites 91.2%

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

   if 1.09999999999999995e-132 < a

   1. Initial program 85.5%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 95.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-PI.f64 90.9

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

   9. Applied rewrites 90.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{\left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right) \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{\left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right) \cdot \left(b - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0245)
   (* (/ (PI) (* a b)) (/ 0.5 a))
   (if (<= a 1.1e-132)
     (/ (* (/ 0.5 b) (PI)) (* a b))
     (/ (- (PI)) (* (* (* (+ a b) 2.0) b) (- b a))))))

Derivation

1. Split input into 3 regimes

2. if a < -0.024500000000000001

   1. Initial program 74.5%

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

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 78.8%

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

   9. Applied rewrites 95.2%

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

   if -0.024500000000000001 < a < 1.09999999999999995e-132

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.4

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

   7. Applied rewrites 68.4%

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

   9. Applied rewrites 91.2%

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

   if 1.09999999999999995e-132 < a

   1. Initial program 85.5%

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

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 95.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

   7. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-PI.f64 90.9

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

   9. Applied rewrites 90.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{PI}\left(\right)}{\left(\left(\left(a + b\right) \cdot 2\right) \cdot b\right) \cdot \left(b - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0245 \lor \neg \left(a \leq 1.16 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.0245) (not (<= a 1.16e-129)))
   (* (/ (PI) (* a b)) (/ 0.5 a))
   (* (PI) (/ 0.5 (* (* b b) a)))))

Derivation

1. Split input into 2 regimes

2. if a < -0.024500000000000001 or 1.16e-129 < a

   1. Initial program 80.8%

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

   3. Taylor expanded in 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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 89.7%

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

   if -0.024500000000000001 < a < 1.16e-129

   1. Initial program 68.7%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.8

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

   7. Applied rewrites 68.8%

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

   9. Applied rewrites 68.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0245 \lor \neg \left(a \leq 1.16 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0245)
   (* (/ (PI) (* a b)) (/ 0.5 a))
   (if (<= a 1.9e-129)
     (/ (* (/ 0.5 b) (PI)) (* a b))
     (/ (* (/ (PI) a) 0.5) (* a b)))))

Derivation

1. Split input into 3 regimes

2. if a < -0.024500000000000001

   1. Initial program 74.5%

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

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 78.8%

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

   9. Applied rewrites 95.2%

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

   if -0.024500000000000001 < a < 1.89999999999999992e-129

   1. Initial program 68.7%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.8

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

   7. Applied rewrites 68.8%

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

   9. Applied rewrites 91.3%

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

   if 1.89999999999999992e-129 < a

   1. Initial program 85.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 85.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{0.5}{b} \cdot \mathsf{PI}\left(\right)}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.16 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0245)
   (* (/ (PI) (* a b)) (/ 0.5 a))
   (if (<= a 1.16e-129)
     (* (PI) (/ 0.5 (* (* b b) a)))
     (/ (* (/ (PI) a) 0.5) (* a b)))))

Derivation

1. Split input into 3 regimes

2. if a < -0.024500000000000001

   1. Initial program 74.5%

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

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   7. Applied rewrites 78.8%

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

   9. Applied rewrites 95.2%

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

   if -0.024500000000000001 < a < 1.16e-129

   1. Initial program 68.7%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.8

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

   7. Applied rewrites 68.8%

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

   9. Applied rewrites 68.8%

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

   if 1.16e-129 < a

   1. Initial program 85.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 85.9%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0245:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot b} \cdot \frac{0.5}{a}\\ \mathbf{elif}\;a \leq 1.16 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a} \cdot 0.5}{a \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0245 \lor \neg \left(a \leq 1.16 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.0245) (not (<= a 1.16e-129)))
   (/ (* (PI) 0.5) (* (* a b) a))
   (* (PI) (/ 0.5 (* (* b b) a)))))

Derivation

1. Split input into 2 regimes

2. if a < -0.024500000000000001 or 1.16e-129 < a

   1. Initial program 80.8%

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

   3. Taylor expanded in 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. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 88.4%

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

   if -0.024500000000000001 < a < 1.16e-129

   1. Initial program 68.7%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   4. Applied rewrites 68.6%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-PI.f64 68.8

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

   7. Applied rewrites 68.8%

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

   9. Applied rewrites 68.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0245 \lor \neg \left(a \leq 1.16 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right) \cdot 0.5}{\left(a \cdot b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{PI}\left(\right) \cdot \frac{0.5}{\left(b \cdot b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.3%

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

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 62.2

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

5. Applied rewrites 62.2%

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

7. Applied rewrites 61.8%

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

9. Applied rewrites 67.9%

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

10. Final simplification 67.9%

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

Alternative 13: 56.9% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.3%

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

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 62.2

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

5. Applied rewrites 62.2%

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

7. Applied rewrites 61.8%

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

8. Final simplification 61.8%

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

Reproduce

herbie shell --seed 2024364 
(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))))