Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 2.3s

Alternatives: 6

Speedup: 1.8×

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, 0.9× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ 4.0 (fma (* (* v v) (PI)) -3.0 (* (PI) 3.0)))
  (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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

7. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

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

   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. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 4.0 (* (PI) 3.0)) (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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 97.6

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

5. Applied rewrites 97.6%

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

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

7. Applied rewrites 99.1%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

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
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. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/r* N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lift-PI.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 98.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (/ (/ 1.3333333333333333 (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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 97.6

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

5. Applied rewrites 97.6%

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

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift--.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in v around 0

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

10. Applied rewrites 99.1%

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

11. Taylor expanded in v around 0

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

    1. lower-/.f64 N/A

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

    2. lift-PI.f64 99.1

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

13. Applied rewrites 99.1%

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

Alternative 6: 97.4% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lift-PI.f64 97.5

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

5. Applied rewrites 97.5%

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

Reproduce

herbie shell --seed 2025079 
(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)))))))