Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 6.9s

Alternatives: 8

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, 0.3× speedup

Math

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

FPCore

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

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. *-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. lift--.f64 N/A

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

   8. flip-- N/A

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

   9. associate-*r/ N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (- 1.0 (* v v)) (* (* (sqrt (fma -6.0 (* v v) 2.0)) 3.0) (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. *-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. lift--.f64 N/A

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

   8. flip-- N/A

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

   9. associate-*r/ N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lift--.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. pow-pow N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-fma.f64 N/A

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

6. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 N/A

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

   17. pow1/2 N/A

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

   18. lift-sqrt.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 96.5

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

7. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 98.0

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

9. Applied rewrites 98.0%

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

10. Taylor expanded in v around 0

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

    1. associate-*r/ N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. div-add-rev N/A

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

    5. lower-/.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 N/A

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

    10. lower-PI.f64 98.9

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

12. Applied rewrites 98.9%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 5: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 96.5

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

7. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 98.0

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

9. Applied rewrites 98.0%

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

Alternative 6: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

4. Applied rewrites 97.9%

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

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 98.0

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

7. Applied rewrites 98.0%

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

Alternative 7: 99.0% 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 98.5%

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

4. Applied rewrites 97.9%

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

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 97.9

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

7. Applied rewrites 97.9%

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

Alternative 8: 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 98.5%

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

   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 96.4

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

5. Applied rewrites 96.4%

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

Reproduce

herbie shell --seed 2024358 
(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)))))))