NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.5% → 99.3%

Time: 7.9s

Alternatives: 12

Speedup: 2.2×

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: 78.5% 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: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+110}:\\ \;\;\;\;\left(\frac{-1}{b} \cdot \frac{\mathsf{PI}\left(\right)}{a + b}\right) \cdot \frac{0.5}{b - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b - a}{a + b}}{\left(\left(b - a\right) \cdot a\right) \cdot b} \cdot \left(0.5 \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e+110)
   (* (* (/ -1.0 b) (/ (PI) (+ a b))) (/ 0.5 (- b a)))
   (* (/ (/ (- b a) (+ a b)) (* (* (- b a) a) b)) (* 0.5 (PI)))))

Derivation

1. Split input into 2 regimes

2. if a < -2e110

   1. Initial program 60.3%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 99.9%

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

   if -2e110 < a

   1. Initial program 79.4%

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

   4. Applied rewrites 91.5%

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.7

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

   8. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 94.3%

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

Alternative 2: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{-1} \cdot \frac{\mathsf{PI}\left(\right)}{a + b}\right) \cdot \frac{0.5}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-109)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (* (* (pow a -1.0) (/ (PI) (+ a b))) (/ 0.5 (- b a)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999996e-109

   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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.1%

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

   9. Applied rewrites 69.0%

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

   11. Applied rewrites 69.4%

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

   if 2.09999999999999996e-109 < b

   1. Initial program 79.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 90.1

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

   7. Applied rewrites 90.1%

      \[\leadsto \left(\color{blue}{\frac{1}{a}} \cdot \frac{\mathsf{PI}\left(\right)}{a + b}\right) \cdot \frac{0.5}{b - a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{-1} \cdot \frac{\mathsf{PI}\left(\right)}{a + b}\right) \cdot \frac{0.5}{b - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.4× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (/ (* (/ (PI) (+ a b)) (- b a)) (* (- b a) (* (* a b) 2.0))))

Derivation

1. Initial program 75.9%

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

4. Applied rewrites 88.3%

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

6. Applied rewrites 98.6%

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

Alternative 4: 92.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{b}}{a} \cdot \frac{0.5}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-109)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (* (/ (/ (PI) b) a) (/ 0.5 (- b a)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999996e-109

   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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.1%

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

   9. Applied rewrites 69.0%

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

   11. Applied rewrites 69.4%

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

   if 2.09999999999999996e-109 < b

   1. Initial program 79.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 89.9

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

   10. Applied rewrites 89.9%

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

Alternative 5: 92.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{PI}\left(\right)}{a}}{b} \cdot \frac{0.5}{b - a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-109)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (* (/ (/ (PI) a) b) (/ 0.5 (- b a)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999996e-109

   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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.1%

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

   9. Applied rewrites 69.0%

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

   11. Applied rewrites 69.4%

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

   if 2.09999999999999996e-109 < b

   1. Initial program 79.0%

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

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in a around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 89.7

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

   7. Applied rewrites 89.7%

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

Alternative 6: 92.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \mathsf{PI}\left(\right)}{a \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-109)
   (/ (/ (* 0.5 (PI)) (* a b)) a)
   (/ (PI) (* (- b a) (* (* a b) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999996e-109

   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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.1%

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

   9. Applied rewrites 69.0%

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

   11. Applied rewrites 69.4%

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

   if 2.09999999999999996e-109 < b

   1. Initial program 79.0%

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

   4. Applied rewrites 98.4%

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-PI.f64 89.1

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

   9. Applied rewrites 89.1%

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

Alternative 7: 92.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{a} \cdot \frac{0.5}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-109)
   (* (/ (PI) a) (/ 0.5 (* a b)))
   (/ (PI) (* (- b a) (* (* a b) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999996e-109

   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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 69.4%

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

   if 2.09999999999999996e-109 < b

   1. Initial program 79.0%

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

   4. Applied rewrites 98.4%

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-PI.f64 89.1

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

   9. Applied rewrites 89.1%

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

Alternative 8: 91.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b - a\right) \cdot \left(\left(a \cdot b\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-109)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (/ (PI) (* (- b a) (* (* a b) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999996e-109

   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. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 69.0%

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

   if 2.09999999999999996e-109 < b

   1. Initial program 79.0%

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

   4. Applied rewrites 98.4%

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-PI.f64 89.1

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

   9. Applied rewrites 89.1%

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

Alternative 9: 83.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 4e-93)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if b < 3.9999999999999996e-93

   1. Initial program 75.0%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 69.7%

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

   if 3.9999999999999996e-93 < b

   1. Initial program 77.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

Alternative 10: 62.6% accurate, 2.6× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (/ (PI) (* (* a b) a)) 0.5))

Derivation

1. Initial program 75.9%

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

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 52.6

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

5. Applied rewrites 52.6%

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

7. Applied rewrites 61.0%

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

Alternative 11: 62.6% accurate, 2.6× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (PI) (/ 0.5 (* (* b a) a))))

Derivation

1. Initial program 75.9%

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

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 52.6

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

5. Applied rewrites 52.6%

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

7. Applied rewrites 52.6%

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

9. Applied rewrites 61.0%

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

Alternative 12: 56.7% accurate, 2.6× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (PI) (/ 0.5 (* (* a a) b))))

Derivation

1. Initial program 75.9%

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

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

   4. lower-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 52.6

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

5. Applied rewrites 52.6%

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

7. Applied rewrites 52.6%

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

Reproduce

herbie shell --seed 2024317 
(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))))