NMSE Section 6.1 mentioned, B

Percentage Accurate: 78.2% → 99.7%

Time: 8.5s

Alternatives: 6

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: 78.2% 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, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.6%

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

4. Applied rewrites 91.6%

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

6. Applied rewrites 99.1%

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

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 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 68.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(a \cdot b\right) \cdot a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{PI}\left(\right)}{\left(b \cdot b\right) \cdot a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e-60)
   (* (/ (PI) (* (* a b) a)) 0.5)
   (* (/ (PI) (* (* b b) a)) 0.5)))

Derivation

1. Split input into 2 regimes

2. if a < -8.6000000000000001e-60

   1. Initial program 83.0%

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

   3. Taylor expanded in 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 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 85.9%

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

   if -8.6000000000000001e-60 < a

   1. Initial program 82.4%

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

   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 66.2

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

   5. Applied rewrites 66.2%

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

Alternative 3: 99.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.6%

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

4. Applied rewrites 91.6%

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

6. Applied rewrites 99.1%

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

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 99.1

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

9. Applied rewrites 99.1%

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

Alternative 4: 62.3% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.2

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

5. Applied rewrites 57.2%

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

7. Applied rewrites 61.6%

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

Alternative 5: 56.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.2

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

5. Applied rewrites 57.2%

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

6. Final simplification 57.2%

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

Alternative 6: 56.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 82.6%

   \[\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
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 57.2

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

5. Applied rewrites 57.2%

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

7. Applied rewrites 57.2%

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

8. Final simplification 57.2%

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

Reproduce

herbie shell --seed 2024271 
(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))))