Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 6.8s

Alternatives: 6

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-3, v \cdot v, 3\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (sqrt (fma (* v v) -6.0 2.0))) (* (PI) (fma -3.0 (* v v) 3.0))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{4}{\color{blue}{\left(-3 \cdot \left({v}^{2} \cdot \mathsf{PI}\left(\right)\right) + 3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(-3 \cdot {v}^{2}\right) \cdot \mathsf{PI}\left(\right)} + 3 \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-3 \cdot {v}^{2} + 3\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-3 \cdot {v}^{2} + 3\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. lower-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(-3 \cdot {v}^{2} + 3\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{{v}^{2} \cdot -3} + 3\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left({v}^{2}, -3, 3\right)}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. unpow2 N/A

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\color{blue}{v \cdot v}, -3, 3\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   8. lower-*.f64 98.5

      \[\leadsto \frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\color{blue}{v \cdot v}, -3, 3\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

5. Applied rewrites 98.5%

   \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{4}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}{\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}{\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(v \cdot v, -3, 3\right)}} \]

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\mathsf{fma}\left(-3, v \cdot v, 3\right) \cdot \mathsf{PI}\left(\right)}} \]

8. Final simplification 100.0%

   \[\leadsto \frac{\frac{4}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-3, v \cdot v, 3\right)} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  1.3333333333333333
  (* (* (- 1.0 (* v v)) (PI)) (sqrt (fma (* v v) -6.0 2.0)))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. flip3-- N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\frac{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}} \]

   3. clear-num N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}}}}} \]

   5. clear-num N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{\frac{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}}} \]

   6. flip3-- N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}}}} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}}}} \]

   8. lower-/.f64 98.5

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1}{2 - 6 \cdot \left(v \cdot v\right)}}}}} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}}}} \]

   10. sub-neg N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{2 + \left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right)}}}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(6 \cdot \left(v \cdot v\right)\right)\right) + 2}}}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\left(\mathsf{neg}\left(\color{blue}{6 \cdot \left(v \cdot v\right)}\right)\right) + 2}}}} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(6\right)\right) \cdot \left(v \cdot v\right)} + 2}}}} \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(6\right), v \cdot v, 2\right)}}}}} \]

   15. metadata-eval 98.5

       \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(\color{blue}{-6}, v \cdot v, 2\right)}}}} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}} \]

   5. lift-PI.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - v \cdot v\right)}\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{v \cdot v}\right)\right) \cdot \sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}} \]

   8. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}}} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}}} \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}}} \]

   11. remove-double-div N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

   12. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{-6 \cdot \left(v \cdot v\right) + 2}}} \]

   13. +-commutative N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + -6 \cdot \left(v \cdot v\right)}}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(\mathsf{neg}\left(6\right)\right)} \cdot \left(v \cdot v\right)}} \]

   15. cancel-sign-sub-inv N/A

       \[\leadsto \frac{\frac{4}{\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
7. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4}{\mathsf{PI}\left(\right) \cdot 3}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* (PI) 3.0)) (sqrt (fma (* v v) -6.0 2.0))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. lower-PI.f64 98.0

      \[\leadsto \frac{4}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

5. Applied rewrites 98.0%

   \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{4}{\left(\mathsf{PI}\left(\right) \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 3\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\mathsf{PI}\left(\right) \cdot 3}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{\mathsf{PI}\left(\right) \cdot 3}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   5. lower-/.f64 99.6

      \[\leadsto \frac{\color{blue}{\frac{4}{\mathsf{PI}\left(\right) \cdot 3}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. lift--.f64 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{2 - \color{blue}{6 \cdot \left(v \cdot v\right)}}} \]

   8. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{2 + \left(\mathsf{neg}\left(6\right)\right) \cdot \left(v \cdot v\right)}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{2 + \color{blue}{-6} \cdot \left(v \cdot v\right)}} \]

   10. +-commutative N/A

       \[\leadsto \frac{\frac{4}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot \mathsf{PI}\left(\right)\right)\right)}}{\sqrt{\color{blue}{-6 \cdot \left(v \cdot v\right) + 2}}} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\frac{4}{3 \cdot \mathsf{PI}\left(\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{4}{\mathsf{PI}\left(\right) \cdot 3}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]
9. Add Preprocessing

Alternative 4: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 1.3333333333333333 (* (sqrt (fma -6.0 (* v v) 2.0)) (PI))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\left(\mathsf{fma}\left(v, v, 1\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]
6. Step-by-step derivation

   1. lower-PI.f64 99.6

      \[\leadsto \frac{1.3333333333333333}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

7. Applied rewrites 99.6%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}} \]

8. Final simplification 99.6%

   \[\leadsto \frac{1.3333333333333333}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)} \cdot \mathsf{PI}\left(\right)} \]
9. Add Preprocessing

Alternative 5: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\sqrt{2} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ 1.3333333333333333 (* (sqrt 2.0) (PI))))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\left(\mathsf{fma}\left(v, v, 1\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}} \]

5. Taylor expanded in v around 0

   \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\sqrt{2}} \cdot \mathsf{PI}\left(\right)} \]

   4. lower-PI.f64 99.5

      \[\leadsto \frac{1.3333333333333333}{\sqrt{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 99.5%

   \[\leadsto \frac{1.3333333333333333}{\color{blue}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}} \]
8. Add Preprocessing

Alternative 6: 97.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (/ (sqrt 0.5) (PI)) 1.3333333333333333))

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\mathsf{PI}\left(\right)}} \cdot \frac{4}{3} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}}}}{\mathsf{PI}\left(\right)} \cdot \frac{4}{3} \]

   5. lower-PI.f64 98.0

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\mathsf{PI}\left(\right)}} \cdot 1.3333333333333333 \]

5. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 (PI)) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))