NMSE Section 6.1 mentioned, B

Percentage Accurate: 77.8% → 99.7%

Time: 8.7s

Alternatives: 7

Speedup: 2.4×

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: 77.8% 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: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.6%

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

4. Applied rewrites 92.2%

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

6. Applied rewrites 86.4%

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

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.6%

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

4. Applied rewrites 92.2%

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

6. Applied rewrites 86.4%

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

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 99.8

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

9. Applied rewrites 99.8%

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

11. Applied rewrites 99.7%

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

Alternative 3: 89.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.4e-8)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* a b) b)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -3.4e-8

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0

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

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.8%

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

   if -3.4e-8 < a

   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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   6. 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 61.6

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

   7. Applied rewrites 61.6%

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

   9. Applied rewrites 68.8%

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

Alternative 4: 83.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3.4e-8)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -3.4e-8

   1. Initial program 83.5%

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

   3. Taylor expanded in b around 0

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

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.8%

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

   if -3.4e-8 < a

   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 b around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{a \cdot {b}^{2}}} \]
   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 61.6

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

   5. Applied rewrites 61.6%

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

Alternative 5: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.6%

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

4. Applied rewrites 92.2%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 99.0

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

9. Applied rewrites 99.0%

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

Alternative 6: 62.4% 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 80.6%

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

3. Taylor expanded in b around 0

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

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 59.1

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

5. Applied rewrites 59.1%

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

7. Applied rewrites 62.7%

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

Alternative 7: 56.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 80.6%

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

3. Taylor expanded in b around 0

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

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 59.1

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

5. Applied rewrites 59.1%

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

6. Final simplification 59.1%

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

Reproduce

herbie shell --seed 2024268 
(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))))