NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.9% → 99.6%

Time: 4.6s

Alternatives: 8

Speedup: 1.8×

Specification

Math

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

FPCore

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

Initial Program: 77.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. difference-of-squares N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 87.2

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

4. Applied rewrites 87.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. difference-of-squares N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 87.2

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

4. Applied rewrites 87.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 99.6%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-+.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l/ N/A

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

    7. lower-/.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lift-PI.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. +-commutative N/A

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

    13. lower-+.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 3: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{a}\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{t\_0}{b \cdot a} \cdot 0.5\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot 0.5}{b}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (PI) a)))
   (if (<= a -1.6e-23)
     (* (/ t_0 (* b a)) 0.5)
     (if (<= a 1.9e-33)
       (/ (/ (* t_0 0.5) b) b)
       (* (/ (PI) (* a (* a b))) 0.5)))))

Derivation

1. Split input into 3 regimes

2. if a < -1.59999999999999988e-23

   1. Initial program 67.5%

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 93.4%

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

   if -1.59999999999999988e-23 < a < 1.89999999999999997e-33

   1. Initial program 81.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 86.8%

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

   if 1.89999999999999997e-33 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 88.6%

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

Alternative 4: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. difference-of-squares N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 87.2

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

4. Applied rewrites 87.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.6%

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

Alternative 5: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23} \lor \neg \left(a \leq 1.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.6e-23) (not (<= a 1.9e-33)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* b (* b a))) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.59999999999999988e-23 or 1.89999999999999997e-33 < a

   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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 90.3%

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

   if -1.59999999999999988e-23 < a < 1.89999999999999997e-33

   1. Initial program 81.3%

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 32.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 74.0%

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

   12. Applied rewrites 85.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23} \lor \neg \left(a \leq 1.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23} \lor \neg \left(a \leq 2.15 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.6e-23) (not (<= a 2.15e-75)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -1.59999999999999988e-23 or 2.15e-75 < a

   1. Initial program 74.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 88.4%

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

   if -1.59999999999999988e-23 < a < 2.15e-75

   1. Initial program 81.3%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 75.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23} \lor \neg \left(a \leq 2.15 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.6e-23)
   (* (/ (/ (PI) a) (* b a)) 0.5)
   (if (<= a 1.9e-33)
     (* (/ (PI) (* b (* b a))) 0.5)
     (* (/ (PI) (* a (* a b))) 0.5))))

Derivation

1. Split input into 3 regimes

2. if a < -1.59999999999999988e-23

   1. Initial program 67.5%

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 93.4%

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

   if -1.59999999999999988e-23 < a < 1.89999999999999997e-33

   1. Initial program 81.3%

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 32.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 74.0%

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

   12. Applied rewrites 85.8%

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

   if 1.89999999999999997e-33 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 88.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.5%

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

5. Applied rewrites 55.6%

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

7. Applied rewrites 61.8%

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

Reproduce

herbie shell --seed 2025026 
(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))))