Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 4.3s

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

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. 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-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 100.0

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. 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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.5

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

5. Applied rewrites 97.7%

   \[\leadsto \frac{4}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 3\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. *-commutative N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. lift-fma.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

7. Applied rewrites 99.2%

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

Alternative 4: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. 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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.5

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 99.2%

   \[\leadsto \frac{1.3333333333333333}{\sqrt{\color{blue}{2}} \cdot \left(\left(1 - v \cdot v\right) \cdot \mathsf{PI}\left(\right)\right)} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 99.2

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

11. Applied rewrites 99.2%

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

Alternative 5: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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. 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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.5

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 100.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 99.2%

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

Alternative 6: 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 98.5%

   \[\frac{4}{\left(\left(3 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
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 97.6%

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

7. Applied rewrites 99.2%

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

Reproduce

herbie shell --seed 2025019 
(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)))))))