NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.8% → 99.6%

Time: 4.6s

Alternatives: 6

Speedup: 1.8×

Specification

Math

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

FPCore

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

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.0%

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

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

   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 85.9

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

4. Applied rewrites 85.9%

   \[\leadsto \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right)}{2}}{b + a} \cdot \frac{1}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 2: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right) \cdot b}{\left(a \cdot b\right) \cdot \left(2 \cdot \left(\left(b + a\right) \cdot b\right)\right)}\\ t_1 := \frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \mathbf{if}\;b \leq -3.25 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -9.8 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (* (PI) b) (* (* a b) (* 2.0 (* (+ b a) b)))))
        (t_1 (/ (* 0.5 (PI)) (* b (* b a)))))
   (if (<= b -3.25e+91)
     t_1
     (if (<= b -9.8e-95)
       t_0
       (if (<= b 6.4e-242)
         (* (/ (/ (PI) a) (* b a)) 0.5)
         (if (<= b 1.35e+70) t_0 t_1))))))

Derivation

1. Split input into 3 regimes

2. if b < -3.2499999999999999e91 or 1.35e70 < b

   1. Initial program 57.8%

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 44.0%

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

   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 75.4%

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

   12. Applied rewrites 98.2%

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

   if -3.2499999999999999e91 < b < -9.8e-95 or 6.39999999999999997e-242 < b < 1.35e70

   1. Initial program 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. frac-sub N/A

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

      14. frac-times N/A

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

      15. frac-times N/A

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.0%

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

   if -9.8e-95 < b < 6.39999999999999997e-242

   1. Initial program 79.8%

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 92.7%

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

   9. Applied rewrites 94.6%

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

Alternative 3: 83.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-51} \lor \neg \left(a \leq 4.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.7e-51) (not (<= a 4.2e-84)))
   (* (/ (/ (PI) a) (* b a)) 0.5)
   (/ (* 0.5 (PI)) (* b (* b a)))))

Derivation

1. Split input into 2 regimes

2. if a < -4.6999999999999997e-51 or 4.19999999999999996e-84 < a

   1. Initial program 78.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.9%

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

   7. Applied rewrites 86.0%

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

   9. Applied rewrites 86.7%

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

   if -4.6999999999999997e-51 < a < 4.19999999999999996e-84

   1. Initial program 72.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 26.1%

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

   7. Applied rewrites 26.1%

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

   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 72.6%

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

   12. Applied rewrites 92.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-51} \lor \neg \left(a \leq 4.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-51} \lor \neg \left(a \leq 4.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.7e-51) (not (<= a 4.2e-84)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (/ (* 0.5 (PI)) (* b (* b a)))))

Derivation

1. Split input into 2 regimes

2. if a < -4.6999999999999997e-51 or 4.19999999999999996e-84 < a

   1. Initial program 78.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.9%

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

   7. Applied rewrites 86.0%

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

   if -4.6999999999999997e-51 < a < 4.19999999999999996e-84

   1. Initial program 72.5%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 26.1%

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

   7. Applied rewrites 26.1%

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

   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 72.6%

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

   12. Applied rewrites 92.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-51} \lor \neg \left(a \leq 4.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{PI}\left(\right)}{b \cdot \left(b \cdot a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-51} \lor \neg \left(a \leq 3.4 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.7e-51) (not (<= a 3.4e-84)))
   (* (/ (PI) (* a (* a b))) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -4.6999999999999997e-51 or 3.40000000000000021e-84 < a

   1. Initial program 78.2%

      \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
   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 75.9%

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

   7. Applied rewrites 86.0%

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

   if -4.6999999999999997e-51 < a < 3.40000000000000021e-84

   1. Initial program 72.5%

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

   3. Taylor expanded in a around 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 72.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-51} \lor \neg \left(a \leq 3.4 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a \cdot \left(a \cdot b\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.8% 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 76.0%

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

3. 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 56.7%

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

7. Applied rewrites 62.8%

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

Reproduce

herbie shell --seed 2025025 
(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))))