Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 4.7s

Alternatives: 5

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

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

FPCore

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

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 100.0

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

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

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

   16. +-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 100.0%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (PI)) (* (sqrt (fma -6.0 (* v v) 2.0)) 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.7

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

9. Applied rewrites 98.7%

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

Alternative 4: 98.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ 1.3333333333333333 (* (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

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 97.2

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

9. Applied rewrites 97.2%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

    6. metadata-eval 98.7

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

    7. lift-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f64 98.7

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

11. Applied rewrites 98.7%

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

Alternative 5: 98.9% accurate, 2.1× speedup

Math

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

FPCore

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

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

5. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\mathsf{PI}\left(\right)} \cdot 1.3333333333333333} \]
6. Step-by-step derivation

7. Applied rewrites 98.7%

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

Reproduce

herbie shell --seed 2025025 
(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)))))))